Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 99.1%

Time: 4.2s

Alternatives: 13

Speedup: 2.6×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.7% accurate, 1.0× speedup

Math

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, a \cdot \mathsf{fma}\left(a, a, a\right)\right), 4, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
         1.0)
        INFINITY)
     (fma
      t_0
      t_0
      (fma (fma (* (fma -3.0 a 1.0) b) b (* a (fma a a a))) 4.0 -1.0))
     (sqrt (pow a 8.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= ((double) INFINITY)) {
		tmp = fma(t_0, t_0, fma(fma((fma(-3.0, a, 1.0) * b), b, (a * fma(a, a, a))), 4.0, -1.0));
	} else {
		tmp = sqrt(pow(a, 8.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= Inf)
		tmp = fma(t_0, t_0, fma(fma(Float64(fma(-3.0, a, 1.0) * b), b, Float64(a * fma(a, a, a))), 4.0, -1.0));
	else
		tmp = sqrt((a ^ 8.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(t$95$0 * t$95$0 + N[(N[(N[(N[(-3.0 * a + 1.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(a * N[(a * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   3. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, a \cdot \mathsf{fma}\left(a, a, a\right)\right), 4, -1\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, {a}^{2} \cdot \left(a - -1\right), {a}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (sqrt (pow a 8.0))
   (if (<= a 5.4e-6)
     (- (fma 4.0 (* b b) (pow b 4.0)) 1.0)
     (- (fma 4.0 (* (pow a 2.0) (- a -1.0)) (pow a 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = sqrt(pow(a, 8.0));
	} else if (a <= 5.4e-6) {
		tmp = fma(4.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma(4.0, (pow(a, 2.0) * (a - -1.0)), pow(a, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = sqrt((a ^ 8.0));
	elseif (a <= 5.4e-6)
		tmp = Float64(fma(4.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(fma(4.0, Float64((a ^ 2.0) * Float64(a - -1.0)), (a ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 5.4e-6], N[(N[(4.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]

   if -3.8000000000000003e32 < a < 5.39999999999999997e-6

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

      3. lower-*.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

   if 5.39999999999999997e-6 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(a + \color{blue}{1}\right), {a}^{4}\right) - 1 \]

      3. add-flip N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(a - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), {a}^{4}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(a - -1\right), {a}^{4}\right) - 1 \]

      5. lower--.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(a - \color{blue}{-1}\right), {a}^{4}\right) - 1 \]

   6. Applied rewrites 53.5%

      \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(a - \color{blue}{-1}\right), {a}^{4}\right) - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (sqrt (pow a 8.0))
   (if (<= a 5.4e-6)
     (- (fma 4.0 (* b b) (pow b 4.0)) 1.0)
     (- (fma (fma a a a) (* 4.0 a) (* (* (* a a) a) a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = sqrt(pow(a, 8.0));
	} else if (a <= 5.4e-6) {
		tmp = fma(4.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma(fma(a, a, a), (4.0 * a), (((a * a) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = sqrt((a ^ 8.0));
	elseif (a <= 5.4e-6)
		tmp = Float64(fma(4.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(fma(fma(a, a, a), Float64(4.0 * a), Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 5.4e-6], N[(N[(4.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]

   if -3.8000000000000003e32 < a < 5.39999999999999997e-6

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

      3. lower-*.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \mathsf{fma}\left(4, b \cdot \color{blue}{b}, {b}^{4}\right) - 1 \]

   if 5.39999999999999997e-6 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)} \cdot 4\right) - 1 \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \color{blue}{\left(a \cdot a\right)} \cdot \left(a \cdot a\right)\right) - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \left(\color{blue}{a} \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      6. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(a, a, a\right) \cdot \left(a \cdot 4\right) + \color{blue}{\left(a \cdot a\right)} \cdot \left(a \cdot a\right)\right) - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), \color{blue}{a \cdot 4}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot \color{blue}{a}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      9. lower-*.f64 57.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot \color{blue}{a}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      14. lower-*.f64 57.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   8. Applied rewrites 57.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), \color{blue}{4 \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (sqrt (pow a 8.0))
   (if (<= a 5.4e-6)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (- (fma (fma a a a) (* 4.0 a) (* (* (* a a) a) a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = sqrt(pow(a, 8.0));
	} else if (a <= 5.4e-6) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = fma(fma(a, a, a), (4.0 * a), (((a * a) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = sqrt((a ^ 8.0));
	elseif (a <= 5.4e-6)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(fma(fma(a, a, a), Float64(4.0 * a), Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 5.4e-6], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]

   if -3.8000000000000003e32 < a < 5.39999999999999997e-6

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 4 \cdot {b}^{2}\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot {b}^{2}\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot b}, 4 \cdot {b}^{2}\right) - 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b} \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      15. lower-*.f64 69.6

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right)} - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{4}\right) - 1 \]

      3. distribute-lft-out N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 4\right)} - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 4\right) - 1 \]

      5. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      6. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b + 4\right)}\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6

         \[\leadsto b \cdot \left(b \cdot \mathsf{fma}\left(b, \color{blue}{b}, 4\right)\right) - 1 \]

   8. Applied rewrites 69.6%

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]

   if 5.39999999999999997e-6 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)} \cdot 4\right) - 1 \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \color{blue}{\left(a \cdot a\right)} \cdot \left(a \cdot a\right)\right) - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4 + \left(\color{blue}{a} \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      6. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(a, a, a\right) \cdot \left(a \cdot 4\right) + \color{blue}{\left(a \cdot a\right)} \cdot \left(a \cdot a\right)\right) - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), \color{blue}{a \cdot 4}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot \color{blue}{a}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      9. lower-*.f64 57.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot \color{blue}{a}, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      14. lower-*.f64 57.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   8. Applied rewrites 57.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), \color{blue}{4 \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (sqrt (pow a 8.0))
   (if (<= a 5.4e-6)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (- (fma (* a a) (* a a) (* (* (fma a a a) a) 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = sqrt(pow(a, 8.0));
	} else if (a <= 5.4e-6) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = fma((a * a), (a * a), ((fma(a, a, a) * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = sqrt((a ^ 8.0));
	elseif (a <= 5.4e-6)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(fma(Float64(a * a), Float64(a * a), Float64(Float64(fma(a, a, a) * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 5.4e-6], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]

   if -3.8000000000000003e32 < a < 5.39999999999999997e-6

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 4 \cdot {b}^{2}\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot {b}^{2}\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot b}, 4 \cdot {b}^{2}\right) - 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b} \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      15. lower-*.f64 69.6

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right)} - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{4}\right) - 1 \]

      3. distribute-lft-out N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 4\right)} - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 4\right) - 1 \]

      5. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      6. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b + 4\right)}\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6

         \[\leadsto b \cdot \left(b \cdot \mathsf{fma}\left(b, \color{blue}{b}, 4\right)\right) - 1 \]

   8. Applied rewrites 69.6%

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]

   if 5.39999999999999997e-6 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{{a}^{8}}\\ \mathbf{elif}\;a \leq 0.9:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (sqrt (pow a 8.0))
   (if (<= a 0.9)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (* (* (* (- a -4.0) a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = sqrt(pow(a, 8.0));
	} else if (a <= 0.9) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (((a - -4.0) * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = sqrt((a ^ 8.0));
	elseif (a <= 0.9)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 0.9], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \color{blue}{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      6. lower-unsound-*.f32 43.5

         \[\leadsto \sqrt{\left( \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \cdot \left( \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \right)_{\text{binary64}} \right)_{\text{binary32}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}^{2}} \]

      10. pow-prod-down N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{2} \cdot {\left(a \cdot a\right)}^{2}} \]

      11. pow-prod-up N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{\left(2 + 2\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{{\left(a \cdot a\right)}^{4}} \]

      14. pow-prod-down N/A

          \[\leadsto \sqrt{{a}^{4} \cdot {a}^{4}} \]

      15. pow-prod-up N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt{{a}^{\left(4 + 4\right)}} \]

      17. metadata-eval 44.0

          \[\leadsto \sqrt{{a}^{8}} \]

   8. Applied rewrites 44.0%

      \[\leadsto \sqrt{{a}^{8}} \]

   if -3.8000000000000003e32 < a < 0.900000000000000022

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 4 \cdot {b}^{2}\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot {b}^{2}\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot b}, 4 \cdot {b}^{2}\right) - 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b} \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      15. lower-*.f64 69.6

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right)} - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{4}\right) - 1 \]

      3. distribute-lft-out N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 4\right)} - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 4\right) - 1 \]

      5. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      6. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b + 4\right)}\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6

         \[\leadsto b \cdot \left(b \cdot \mathsf{fma}\left(b, \color{blue}{b}, 4\right)\right) - 1 \]

   8. Applied rewrites 69.6%

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]

   if 0.900000000000000022 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \color{blue}{\frac{1}{a}}\right) \]

      5. lower-/.f64 45.5

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{\color{blue}{a}}\right) \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto {a}^{4} \cdot 1 + \color{blue}{{a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      6. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + 2\right)} + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      7. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      8. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {a}^{2} + {\color{blue}{a}}^{4} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      9. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(\frac{1}{a} \cdot 4\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot {a}^{-1}\right) \cdot 4\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{\left(4 + -1\right)} \cdot 4\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{3} \cdot 4\right) \]

      21. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      25. lift-*.f64 29.2

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

   6. Applied rewrites 29.2%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot 4} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot 4 \]

      4. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]

      5. distribute-lft-out N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \color{blue}{a} \cdot 4\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, \color{blue}{a}, a \cdot 4\right) \]

      9. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

      10. lower-*.f64 45.4

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, a, 4 \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(a \cdot a\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \left(a \cdot \left(a + 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(a + 4\right) \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      7. sum-to-mult N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      9. lower-unsound-+.f32 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      10. lower-unsound-/.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + 1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      14. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      15. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      17. associate-*r* N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      20. associate-*r* N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{elif}\;a \leq 0.9:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+32)
   (* (* (* a a) a) a)
   (if (<= a 0.9)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (* (* (* (- a -4.0) a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+32) {
		tmp = ((a * a) * a) * a;
	} else if (a <= 0.9) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (((a - -4.0) * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+32)
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	elseif (a <= 0.9)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+32], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 0.9], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000003e32

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      5. lower-*.f64 45.1

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

   8. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a} \]

   if -3.8000000000000003e32 < a < 0.900000000000000022

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.6

         \[\leadsto \mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 4 \cdot {b}^{2}\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 4 \cdot {b}^{2}\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot {b}^{2}\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot b}, 4 \cdot {b}^{2}\right) - 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b} \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot {b}^{2}\right) - 1 \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, 4 \cdot \left(b \cdot b\right)\right) - 1 \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

      15. lower-*.f64 69.6

          \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right)} - 1 \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 4}\right) - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{4}\right) - 1 \]

      3. distribute-lft-out N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 4\right)} - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 4\right) - 1 \]

      5. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      6. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 4\right)\right)} - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b + 4\right)}\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6

         \[\leadsto b \cdot \left(b \cdot \mathsf{fma}\left(b, \color{blue}{b}, 4\right)\right) - 1 \]

   8. Applied rewrites 69.6%

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]

   if 0.900000000000000022 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \color{blue}{\frac{1}{a}}\right) \]

      5. lower-/.f64 45.5

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{\color{blue}{a}}\right) \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto {a}^{4} \cdot 1 + \color{blue}{{a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      6. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + 2\right)} + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      7. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      8. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {a}^{2} + {\color{blue}{a}}^{4} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      9. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(\frac{1}{a} \cdot 4\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot {a}^{-1}\right) \cdot 4\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{\left(4 + -1\right)} \cdot 4\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{3} \cdot 4\right) \]

      21. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      25. lift-*.f64 29.2

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

   6. Applied rewrites 29.2%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot 4} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot 4 \]

      4. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]

      5. distribute-lft-out N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \color{blue}{a} \cdot 4\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, \color{blue}{a}, a \cdot 4\right) \]

      9. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

      10. lower-*.f64 45.4

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, a, 4 \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(a \cdot a\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \left(a \cdot \left(a + 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(a + 4\right) \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      7. sum-to-mult N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      9. lower-unsound-+.f32 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      10. lower-unsound-/.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + 1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      14. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      15. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      17. associate-*r* N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      20. associate-*r* N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   (- (* (* a a) 4.0) 1.0)
   (* (* (* (- a -4.0) a) a) a)))

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (((a - -4.0) * a) * a) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (((a - (-4.0d0)) * a) * a) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (((a - -4.0) * a) * a) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (((a - -4.0) * a) * a) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (((a - -4.0) * a) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

      2. lower-pow.f64 52.0

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

   9. Applied rewrites 52.0%

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

       2. lift-pow.f64 N/A

          \[\leadsto 4 \cdot {a}^{2} - 1 \]

       3. pow2 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       4. lift-*.f64 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       5. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

       6. lower-*.f64 52.0

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \color{blue}{\frac{1}{a}}\right) \]

      5. lower-/.f64 45.5

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{\color{blue}{a}}\right) \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto {a}^{4} \cdot 1 + \color{blue}{{a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      6. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + 2\right)} + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      7. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      8. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {a}^{2} + {\color{blue}{a}}^{4} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      9. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(\frac{1}{a} \cdot 4\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot {a}^{-1}\right) \cdot 4\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{\left(4 + -1\right)} \cdot 4\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{3} \cdot 4\right) \]

      21. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      25. lift-*.f64 29.2

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

   6. Applied rewrites 29.2%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot 4} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot 4 \]

      4. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]

      5. distribute-lft-out N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \color{blue}{a} \cdot 4\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, \color{blue}{a}, a \cdot 4\right) \]

      9. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

      10. lower-*.f64 45.4

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, a, 4 \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a + 4 \cdot a\right) \cdot \left(a \cdot a\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \left(a \cdot \left(a + 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(a + 4\right) \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      7. sum-to-mult N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      9. lower-unsound-+.f32 N/A

         \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      10. lower-unsound-/.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \left(\left(\left(1 + \frac{4}{a}\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + 1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      14. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      15. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) \]

      17. associate-*r* N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      20. associate-*r* N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \color{blue}{a} \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(a - -4\right)\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   (- (* (* a a) 4.0) 1.0)
   (* (* (* a a) a) (- a -4.0))))

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * (a - -4.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((a * a) * a) * (a - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * (a - -4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((a * a) * a) * (a - -4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * Float64(a - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((a * a) * a) * (a - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(a - -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

      2. lower-pow.f64 52.0

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

   9. Applied rewrites 52.0%

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

       2. lift-pow.f64 N/A

          \[\leadsto 4 \cdot {a}^{2} - 1 \]

       3. pow2 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       4. lift-*.f64 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       5. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

       6. lower-*.f64 52.0

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \color{blue}{\frac{1}{a}}\right) \]

      5. lower-/.f64 45.5

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{\color{blue}{a}}\right) \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto {a}^{4} \cdot 1 + \color{blue}{{a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      6. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + 2\right)} + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      7. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      8. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {a}^{2} + {\color{blue}{a}}^{4} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      9. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(\frac{1}{a} \cdot 4\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot {a}^{-1}\right) \cdot 4\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{\left(4 + -1\right)} \cdot 4\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{3} \cdot 4\right) \]

      21. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      25. lift-*.f64 29.2

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

   6. Applied rewrites 29.2%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot 4} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot \color{blue}{a}\right) \cdot 4 \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot 4 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a + \left(\color{blue}{\left(a \cdot a\right)} \cdot a\right) \cdot 4 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a + \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{4} \]

      6. distribute-lft-out N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{\left(a + 4\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{\left(a + 4\right)} \]

      8. add-flip N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(a - \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(a - \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right) \]

      10. metadata-eval 45.4

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(a - -4\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{\left(a - -4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq 10:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
       1.0)
      10.0)
   (- (* (* a a) 4.0) 1.0)
   (* (* (* a a) a) a)))

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= 10.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((a * a) * a) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((a * a) * a) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= 10.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((a * a) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 10.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < 10

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

      2. lower-pow.f64 52.0

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

   9. Applied rewrites 52.0%

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

       2. lift-pow.f64 N/A

          \[\leadsto 4 \cdot {a}^{2} - 1 \]

       3. pow2 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       4. lift-*.f64 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       5. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

       6. lower-*.f64 52.0

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   if 10 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      5. lower-*.f64 45.1

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

   8. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq 10:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
       1.0)
      10.0)
   (- (* (* a a) 4.0) 1.0)
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= 10.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= 10.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= 10.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 10.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < 10

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

      2. lower-pow.f64 52.0

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

   9. Applied rewrites 52.0%

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

       2. lift-pow.f64 N/A

          \[\leadsto 4 \cdot {a}^{2} - 1 \]

       3. pow2 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       4. lift-*.f64 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       5. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

       6. lower-*.f64 52.0

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   if 10 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 45.1

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{a} \cdot a\right) \]

      8. lift-*.f64 45.0

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

   6. Applied rewrites 45.0%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.5) (- (* (* a a) 4.0) 1.0) (* (* a a) (* 4.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (4.0 * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 0.5d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (4.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (4.0 * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (a * a) * (4.0 * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(4.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (4.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.5

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

      4. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

      5. lower-pow.f64 53.5

         \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      5. pow-prod-up N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      6. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      7. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      13. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      15. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      17. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      20. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      21. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

      22. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

      23. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

      24. lower-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   6. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

      2. lower-pow.f64 52.0

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

   9. Applied rewrites 52.0%

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

       2. lift-pow.f64 N/A

          \[\leadsto 4 \cdot {a}^{2} - 1 \]

       3. pow2 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       4. lift-*.f64 N/A

          \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

       5. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

       6. lower-*.f64 52.0

          \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

   if 0.5 < a

   1. Initial program 73.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \color{blue}{\frac{1}{a}}\right) \]

      5. lower-/.f64 45.5

         \[\leadsto {a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{\color{blue}{a}}\right) \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 + \color{blue}{4 \cdot \frac{1}{a}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto {a}^{4} \cdot 1 + \color{blue}{{a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto {a}^{4} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      6. metadata-eval N/A

         \[\leadsto {a}^{\left(2 + 2\right)} + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      7. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      8. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {a}^{2} + {\color{blue}{a}}^{4} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      9. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + {a}^{\color{blue}{4}} \cdot \left(4 \cdot \frac{1}{a}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(4 \cdot \frac{1}{a}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{4} \cdot \left(\frac{1}{a} \cdot 4\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot \frac{1}{a}\right) \cdot 4\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{4} \cdot {a}^{-1}\right) \cdot 4\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{\left(4 + -1\right)} \cdot 4\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, {a}^{3} \cdot 4\right) \]

      21. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(a \cdot \left(a \cdot a\right)\right) \cdot 4\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

      25. lift-*.f64 29.2

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]

   6. Applied rewrites 29.2%

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\left(a \cdot a\right) \cdot a\right) \cdot 4\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot 4} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{4} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(a \cdot a\right) \cdot a\right) \cdot 4 \]

      4. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]

      5. distribute-lft-out N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a + a \cdot 4\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \color{blue}{a} \cdot 4\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, \color{blue}{a}, a \cdot 4\right) \]

      9. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

      10. lower-*.f64 45.4

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot a\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(a \cdot a\right) \cdot \left(4 \cdot \color{blue}{a}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 19.0

          \[\leadsto \left(a \cdot a\right) \cdot \left(4 \cdot a\right) \]

   11. Applied rewrites 19.0%

       \[\leadsto \left(a \cdot a\right) \cdot \left(4 \cdot \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.0% accurate, 5.9× speedup

Math

\[\left(a \cdot a\right) \cdot 4 - 1 \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* a a) 4.0) 1.0))

C

double code(double a, double b) {
	return ((a * a) * 4.0) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((a * a) * 4.0d0) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((a * a) * 4.0) - 1.0;
}

Python

def code(a, b):
	return ((a * a) * 4.0) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(a * a) * 4.0) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((a * a) * 4.0) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 73.7%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

2. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{{a}^{2} \cdot \left(1 + a\right)}, {a}^{4}\right) - 1 \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \color{blue}{\left(1 + a\right)}, {a}^{4}\right) - 1 \]

   3. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(\color{blue}{1} + a\right), {a}^{4}\right) - 1 \]

   4. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + \color{blue}{a}\right), {a}^{4}\right) - 1 \]

   5. lower-pow.f64 53.5

      \[\leadsto \mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right) - 1 \]

4. Applied rewrites 53.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \color{blue}{{a}^{4}}\right) - 1 \]

   2. +-commutative N/A

      \[\leadsto \left({a}^{4} + \color{blue}{4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)}\right) - 1 \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({a}^{4} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   4. metadata-eval N/A

      \[\leadsto \left({a}^{\left(2 + 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   5. pow-prod-up N/A

      \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{4} \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   6. pow2 N/A

      \[\leadsto \left(\left(a \cdot a\right) \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   7. pow2 N/A

      \[\leadsto \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a} \cdot a, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot \color{blue}{a}, 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1 \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   13. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left({a}^{2} \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   14. pow2 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   15. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   17. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4\right) - 1 \]

   18. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1 \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

   20. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(1 + a\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

   21. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(\left(a + 1\right) \cdot a\right) \cdot a\right) \cdot 4\right) - 1 \]

   22. distribute-lft1-in N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\left(a \cdot a + a\right) \cdot a\right) \cdot 4\right) - 1 \]

   23. lift-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

   24. lower-*.f64 53.4

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

6. Applied rewrites 53.4%

   \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a}, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1 \]

7. Taylor expanded in a around 0

   \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

   2. lower-pow.f64 52.0

      \[\leadsto 4 \cdot {a}^{2} - 1 \]

9. Applied rewrites 52.0%

   \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 4 \cdot {a}^{\color{blue}{2}} - 1 \]

    2. lift-pow.f64 N/A

       \[\leadsto 4 \cdot {a}^{2} - 1 \]

    3. pow2 N/A

       \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

    4. lift-*.f64 N/A

       \[\leadsto 4 \cdot \left(a \cdot a\right) - 1 \]

    5. *-commutative N/A

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

    6. lower-*.f64 52.0

       \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]

11. Applied rewrites 52.0%

    \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025174 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))