Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * Math.log(t))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return (((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * Math.log(t))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return (((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \left(y + x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (- a 0.5) b (fma (- z) (log t) (+ z (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((a - 0.5), b, fma(-z, log(t), (z + (y + x))));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(a - 0.5), b, fma(Float64(-z), log(t), Float64(z + Float64(y + x))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a - 0.5), $MachinePrecision] * b + N[((-z) * N[Log[t], $MachinePrecision] + N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - \frac{1}{2}\right) \cdot b} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) \]

   4. lower-fma.f64 99.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) - z \cdot \log t}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\left(x + y\right) + z\right) - \color{blue}{z \cdot \log t}\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \log t + \left(\left(x + y\right) + z\right)}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), \log t, \left(x + y\right) + z\right)}\right) \]

   10. lower-neg.f64 99.9

       \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(\color{blue}{-z}, \log t, \left(x + y\right) + z\right)\right) \]

   11. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{\left(x + y\right) + z}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

   13. lower-+.f64 99.9

       \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(x + y\right)}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

   16. lower-+.f64 99.9

       \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \left(y + x\right)\right)\right)} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ t_2 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, t\_1 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))) (t_2 (* (- a 0.5) b)))
   (if (<= t_2 -4e+96)
     (fma (- a 0.5) b (+ y x))
     (if (<= t_2 2e+164) (fma t_1 z (+ y x)) (fma (- a 0.5) b (* t_1 z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double t_2 = (a - 0.5) * b;
	double tmp;
	if (t_2 <= -4e+96) {
		tmp = fma((a - 0.5), b, (y + x));
	} else if (t_2 <= 2e+164) {
		tmp = fma(t_1, z, (y + x));
	} else {
		tmp = fma((a - 0.5), b, (t_1 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	t_2 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_2 <= -4e+96)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	elseif (t_2 <= 2e+164)
		tmp = fma(t_1, z, Float64(y + x));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(t_1 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+96], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+164], N[(t$95$1 * z + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(t$95$1 * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e96

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 92.2

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   if -4.0000000000000002e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2e164

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 70.0

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + y\right) + z\right)} + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \log t\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{-1 \cdot \left(z \cdot \log t\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + \left(z + -1 \cdot \left(z \cdot \log t\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + -1 \cdot \left(z \cdot \log t\right)\right) + \left(x + y\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + x\right)} \]

   if 2e164 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - \frac{1}{2}\right) \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) - z \cdot \log t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\left(x + y\right) + z\right) - \color{blue}{z \cdot \log t}\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \log t + \left(\left(x + y\right) + z\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), \log t, \left(x + y\right) + z\right)}\right) \]

      10. lower-neg.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(\color{blue}{-z}, \log t, \left(x + y\right) + z\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{\left(x + y\right) + z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

      13. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(x + y\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

      16. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \left(y + x\right)\right)\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{x + \left(z + -1 \cdot \left(z \cdot \log t\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(z + -1 \cdot \left(z \cdot \log t\right)\right) + x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(-1 \cdot \left(z \cdot \log t\right) + z\right)} + x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot \log t\right)\right)} + z\right) + x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\left(\mathsf{neg}\left(\color{blue}{\log t \cdot z}\right)\right) + z\right) + x\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(\log t\right)\right) \cdot z} + z\right) + x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(-1 \cdot \log t\right)} \cdot z + z\right) + x\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(-1 \cdot \log t + 1\right) \cdot z} + x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(1 + -1 \cdot \log t\right)} \cdot z + x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{z \cdot \left(1 + -1 \cdot \log t\right)} + x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(1 \cdot z\right)} \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right) \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)} \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \log t\right)} + x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \left(1 - \color{blue}{1} \cdot \log t\right) + x\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \left(1 - \color{blue}{\log t}\right) + x\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right) \cdot \left(1 - \log t\right) + x\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot z\right)} \cdot \left(1 - \log t\right) + x\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{1} \cdot z\right) \cdot \left(1 - \log t\right) + x\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{z} \cdot \left(1 - \log t\right) + x\right) \]

   7. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{\mathsf{fma}\left(1 - \log t, z, x\right)}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, z \cdot \color{blue}{\left(1 - \log t\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

       \[\leadsto \mathsf{fma}\left(a - 0.5, b, \left(1 - \log t\right) \cdot \color{blue}{z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ t_2 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))) (t_2 (* (- a 0.5) b)))
   (if (<= t_2 -4e+96)
     (fma (- a 0.5) b (+ y x))
     (if (<= t_2 2e+164) (fma t_1 z (+ y x)) (fma t_1 z t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double t_2 = (a - 0.5) * b;
	double tmp;
	if (t_2 <= -4e+96) {
		tmp = fma((a - 0.5), b, (y + x));
	} else if (t_2 <= 2e+164) {
		tmp = fma(t_1, z, (y + x));
	} else {
		tmp = fma(t_1, z, t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	t_2 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_2 <= -4e+96)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	elseif (t_2 <= 2e+164)
		tmp = fma(t_1, z, Float64(y + x));
	else
		tmp = fma(t_1, z, t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+96], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+164], N[(t$95$1 * z + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e96

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 92.2

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   if -4.0000000000000002e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2e164

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 70.0

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + y\right) + z\right)} + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \log t\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{-1 \cdot \left(z \cdot \log t\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + \left(z + -1 \cdot \left(z \cdot \log t\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + -1 \cdot \left(z \cdot \log t\right)\right) + \left(x + y\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + x\right)} \]

   if 2e164 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) - z \cdot \log t\right)} \]

      2. +-commutative N/A

         \[\leadsto y + \left(\color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + z\right)} - z \cdot \log t\right) \]

      3. associate--l+ N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + \left(z - z \cdot \log t\right)\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \left(\color{blue}{z \cdot 1} - z \cdot \log t\right)\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{\left(1 - \log t\right) \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)\right)} \]

      9. associate-+r- N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + z \cdot \left(1 - \log t\right)} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right) + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      15. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \color{blue}{\log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + y}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + y\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, y\right)}\right) \]

      19. lower--.f64 95.8

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(\color{blue}{a - 0.5}, b, y\right)\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(1 - \log t, z, b \cdot \left(a - \frac{1}{2}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(1 - \log t, z, \left(a - 0.5\right) \cdot b\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+96} \lor \neg \left(t\_1 \leq 10^{+158}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -4e+96) (not (<= t_1 1e+158)))
     (fma (- a 0.5) b (+ y x))
     (fma (- 1.0 (log t)) z (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -4e+96) || !(t_1 <= 1e+158)) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -4e+96) || !(t_1 <= 1e+158))
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+96], N[Not[LessEqual[t$95$1, 1e+158]], $MachinePrecision]], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e96 or 9.99999999999999953e157 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 92.3

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   if -4.0000000000000002e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.99999999999999953e157

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 69.8

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + y\right) + z\right)} + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \log t\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(x + y\right) + z\right) + \color{blue}{-1 \cdot \left(z \cdot \log t\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + \left(z + -1 \cdot \left(z \cdot \log t\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + -1 \cdot \left(z \cdot \log t\right)\right) + \left(x + y\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -4 \cdot 10^{+96} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 10^{+158}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -5e-127)
   (fma b (- a 0.5) x)
   (fma b (- a 0.5) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + y) + z) - (z * log(t))) <= -5e-127) {
		tmp = fma(b, (a - 0.5), x);
	} else {
		tmp = fma(b, (a - 0.5), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) <= -5e-127)
		tmp = fma(b, Float64(a - 0.5), x);
	else
		tmp = fma(b, Float64(a - 0.5), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-127], N[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision], N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -4.9999999999999997e-127

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 82.9

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{a - 0.5}, x\right) \]

   if -4.9999999999999997e-127 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 76.2

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{a - 0.5}, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(t\_1, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -5e-127)
     (fma (- a 0.5) b (fma t_1 z x))
     (fma t_1 z (fma (- a 0.5) b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((x + y) <= -5e-127) {
		tmp = fma((a - 0.5), b, fma(t_1, z, x));
	} else {
		tmp = fma(t_1, z, fma((a - 0.5), b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(x + y) <= -5e-127)
		tmp = fma(Float64(a - 0.5), b, fma(t_1, z, x));
	else
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -5e-127], N[(N[(a - 0.5), $MachinePrecision] * b + N[(t$95$1 * z + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999997e-127

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - \frac{1}{2}\right) \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) \]

      4. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \left(\left(x + y\right) + z\right) - z \cdot \log t\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) - z \cdot \log t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\left(x + y\right) + z\right) - \color{blue}{z \cdot \log t}\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(x + y\right) + z\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot \log t}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \log t + \left(\left(x + y\right) + z\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), \log t, \left(x + y\right) + z\right)}\right) \]

      10. lower-neg.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(\color{blue}{-z}, \log t, \left(x + y\right) + z\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{\left(x + y\right) + z}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

      13. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, \color{blue}{z + \left(x + y\right)}\right)\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(x + y\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

      16. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \color{blue}{\left(y + x\right)}\right)\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(-z, \log t, z + \left(y + x\right)\right)\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{x + \left(z + -1 \cdot \left(z \cdot \log t\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(z + -1 \cdot \left(z \cdot \log t\right)\right) + x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(-1 \cdot \left(z \cdot \log t\right) + z\right)} + x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot \log t\right)\right)} + z\right) + x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\left(\mathsf{neg}\left(\color{blue}{\log t \cdot z}\right)\right) + z\right) + x\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(\log t\right)\right) \cdot z} + z\right) + x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(-1 \cdot \log t\right)} \cdot z + z\right) + x\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(-1 \cdot \log t + 1\right) \cdot z} + x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(1 + -1 \cdot \log t\right)} \cdot z + x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{z \cdot \left(1 + -1 \cdot \log t\right)} + x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(1 \cdot z\right)} \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right) \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\mathsf{neg}\left(-1 \cdot z\right)\right)} \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) \cdot \left(1 + -1 \cdot \log t\right) + x\right) \]

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \log t\right)} + x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \left(1 - \color{blue}{1} \cdot \log t\right) + x\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot \left(1 - \color{blue}{\log t}\right) + x\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right) \cdot \left(1 - \log t\right) + x\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot z\right)} \cdot \left(1 - \log t\right) + x\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \left(\color{blue}{1} \cdot z\right) \cdot \left(1 - \log t\right) + x\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{z} \cdot \left(1 - \log t\right) + x\right) \]

   7. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{\mathsf{fma}\left(1 - \log t, z, x\right)}\right) \]

   if -4.9999999999999997e-127 < (+.f64 x y)

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) - z \cdot \log t\right)} \]

      2. +-commutative N/A

         \[\leadsto y + \left(\color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + z\right)} - z \cdot \log t\right) \]

      3. associate--l+ N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + \left(z - z \cdot \log t\right)\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \left(\color{blue}{z \cdot 1} - z \cdot \log t\right)\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{\left(1 - \log t\right) \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)\right)} \]

      9. associate-+r- N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + z \cdot \left(1 - \log t\right)} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right) + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      15. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \color{blue}{\log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + y}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + y\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, y\right)}\right) \]

      19. lower--.f64 81.5

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(\color{blue}{a - 0.5}, b, y\right)\right) \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;x + y \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ x y) -5e-127)
     (fma t_1 z (fma (- a 0.5) b x))
     (fma t_1 z (fma (- a 0.5) b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if ((x + y) <= -5e-127) {
		tmp = fma(t_1, z, fma((a - 0.5), b, x));
	} else {
		tmp = fma(t_1, z, fma((a - 0.5), b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(x + y) <= -5e-127)
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, x));
	else
		tmp = fma(t_1, z, fma(Float64(a - 0.5), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -5e-127], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999997e-127

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - \color{blue}{\log t \cdot z} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \left(\mathsf{neg}\left(\log t\right)\right) \cdot z} \]

      3. mul-1-neg N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \color{blue}{\left(-1 \cdot \log t\right)} \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \color{blue}{z \cdot \left(-1 \cdot \log t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + z \cdot \color{blue}{\left(\mathsf{neg}\left(\log t\right)\right)} \]

      6. log-rec N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + z \cdot \color{blue}{\log \left(\frac{1}{t}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \log \left(\frac{1}{t}\right) + \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto z \cdot \log \left(\frac{1}{t}\right) + \color{blue}{\left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) + x\right)} \]

      9. associate-+l+ N/A

         \[\leadsto z \cdot \log \left(\frac{1}{t}\right) + \color{blue}{\left(z + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)\right)} \]

      10. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(z \cdot \log \left(\frac{1}{t}\right) + z\right) + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)} \]

   if -4.9999999999999997e-127 < (+.f64 x y)

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) - z \cdot \log t\right)} \]

      2. +-commutative N/A

         \[\leadsto y + \left(\color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + z\right)} - z \cdot \log t\right) \]

      3. associate--l+ N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + \left(z - z \cdot \log t\right)\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \left(\color{blue}{z \cdot 1} - z \cdot \log t\right)\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{\left(1 - \log t\right) \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)\right)} \]

      9. associate-+r- N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + z \cdot \left(1 - \log t\right)} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right) + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      15. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \color{blue}{\log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + y}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + y\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, y\right)}\right) \]

      19. lower--.f64 81.5

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(\color{blue}{a - 0.5}, b, y\right)\right) \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 3.9e+51)
   (fma (- 1.0 (log t)) z (fma (- a 0.5) b x))
   (fma (- a 0.5) b (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 3.9e+51) {
		tmp = fma((1.0 - log(t)), z, fma((a - 0.5), b, x));
	} else {
		tmp = fma((a - 0.5), b, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 3.9e+51)
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(a - 0.5), b, x));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 3.9e+51], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.89999999999999984e51

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - \color{blue}{\log t \cdot z} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \left(\mathsf{neg}\left(\log t\right)\right) \cdot z} \]

      3. mul-1-neg N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \color{blue}{\left(-1 \cdot \log t\right)} \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + \color{blue}{z \cdot \left(-1 \cdot \log t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + z \cdot \color{blue}{\left(\mathsf{neg}\left(\log t\right)\right)} \]

      6. log-rec N/A

         \[\leadsto \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) + z \cdot \color{blue}{\log \left(\frac{1}{t}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \log \left(\frac{1}{t}\right) + \left(x + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto z \cdot \log \left(\frac{1}{t}\right) + \color{blue}{\left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) + x\right)} \]

      9. associate-+l+ N/A

         \[\leadsto z \cdot \log \left(\frac{1}{t}\right) + \color{blue}{\left(z + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)\right)} \]

      10. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(z \cdot \log \left(\frac{1}{t}\right) + z\right) + \left(b \cdot \left(a - \frac{1}{2}\right) + x\right)} \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, x\right)\right)} \]

   if 3.89999999999999984e51 < y

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 82.8

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 4.9e+148) (fma (- a 0.5) b (+ y x)) (fma (- 1.0 (log t)) z y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4.9e+148) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((1.0 - log(t)), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4.9e+148)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(Float64(1.0 - log(t)), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 4.9e+148], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9e148

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 85.7

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   if 4.9e148 < z

   1. Initial program 99.6%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y + \left(z + b \cdot \left(a - \frac{1}{2}\right)\right)\right) - z \cdot \log t} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(\left(z + b \cdot \left(a - \frac{1}{2}\right)\right) - z \cdot \log t\right)} \]

      2. +-commutative N/A

         \[\leadsto y + \left(\color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + z\right)} - z \cdot \log t\right) \]

      3. associate--l+ N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + \left(z - z \cdot \log t\right)\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \left(\color{blue}{z \cdot 1} - z \cdot \log t\right)\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{\left(1 - \log t\right) \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto y + \left(b \cdot \left(a - \frac{1}{2}\right) + \color{blue}{z \cdot \left(1 - \log t\right)}\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto y + \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)\right)} \]

      9. associate-+r- N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) - \left(\mathsf{neg}\left(z\right)\right) \cdot \left(1 - \log t\right)} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + z \cdot \left(1 - \log t\right)} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right) + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      15. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \color{blue}{\log t}, z, y + b \cdot \left(a - \frac{1}{2}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + y}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + y\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, y\right)}\right) \]

      19. lower--.f64 88.1

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(\color{blue}{a - 0.5}, b, y\right)\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(a - 0.5, b, y\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto y + \color{blue}{z \cdot \left(1 - \log t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.2%

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

Alternative 10: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 2.3e+208) (fma (- a 0.5) b (+ y x)) (* (- 1.0 (log t)) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 2.3e+208) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = (1.0 - log(t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 2.3e+208)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(Float64(1.0 - log(t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 2.3e+208], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.3e208

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 84.4

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   if 2.3e208 < z

   1. Initial program 99.6%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \log t\right)} \cdot z \]

      4. lower-log.f64 77.2

         \[\leadsto \left(1 - \color{blue}{\log t}\right) \cdot z \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(1 - \log t\right) \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 36.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.78 \lor \neg \left(a \leq 4.4 \cdot 10^{-12}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -0.78) (not (<= a 4.4e-12))) (* b a) (* b -0.5)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -0.78) || !(a <= 4.4e-12)) {
		tmp = b * a;
	} else {
		tmp = b * -0.5;
	}
	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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.78d0)) .or. (.not. (a <= 4.4d-12))) then
        tmp = b * a
    else
        tmp = b * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -0.78) || !(a <= 4.4e-12)) {
		tmp = b * a;
	} else {
		tmp = b * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -0.78) or not (a <= 4.4e-12):
		tmp = b * a
	else:
		tmp = b * -0.5
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -0.78) || !(a <= 4.4e-12))
		tmp = Float64(b * a);
	else
		tmp = Float64(b * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -0.78) || ~((a <= 4.4e-12)))
		tmp = b * a;
	else
		tmp = b * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -0.78], N[Not[LessEqual[a, 4.4e-12]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(b * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.78000000000000003 or 4.39999999999999983e-12 < a

   1. Initial program 99.8%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{b \cdot a} \]

      2. lower-*.f64 48.1

         \[\leadsto \color{blue}{b \cdot a} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{b \cdot a} \]

   if -0.78000000000000003 < a < 4.39999999999999983e-12

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

      9. lower-+.f64 80.1

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]

      2. lower--.f64 24.6

         \[\leadsto b \cdot \color{blue}{\left(a - 0.5\right)} \]

   8. Applied rewrites 24.6%

      \[\leadsto \color{blue}{b \cdot \left(a - 0.5\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto b \cdot \frac{-1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.7%

       \[\leadsto b \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.78 \lor \neg \left(a \leq 4.4 \cdot 10^{-12}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a - 0.5, b, y + x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma (- a 0.5) b (+ y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((a - 0.5), b, (y + x));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(a - 0.5), b, Float64(y + x))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

   5. +-commutative N/A

      \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

   9. lower-+.f64 79.5

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

5. Applied rewrites 79.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]
6. Add Preprocessing

Alternative 13: 57.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b, a - 0.5, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma b (- a 0.5) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(b, (a - 0.5), x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(b, Float64(a - 0.5), x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

   5. +-commutative N/A

      \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

   9. lower-+.f64 79.5

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

5. Applied rewrites 79.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto x + \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 57.0%

   \[\leadsto \mathsf{fma}\left(b, \color{blue}{a - 0.5}, x\right) \]
9. Add Preprocessing

Alternative 14: 37.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(a - 0.5\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* b (- a 0.5)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return b * (a - 0.5);
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (a - 0.5d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return b * (a - 0.5);
}

Python

def code(x, y, z, t, a, b):
	return b * (a - 0.5)

Julia

function code(x, y, z, t, a, b)
	return Float64(b * Float64(a - 0.5))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = b * (a - 0.5);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(y + b \cdot \left(a - \frac{1}{2}\right)\right) + x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot \left(a - \frac{1}{2}\right) + y\right)} + x \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right) + \left(y + x\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot b} + \left(y + x\right) \]

   5. +-commutative N/A

      \[\leadsto \left(a - \frac{1}{2}\right) \cdot b + \color{blue}{\left(x + y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, b, x + y\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, b, x + y\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, \color{blue}{y + x}\right) \]

   9. lower-+.f64 79.5

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{y + x}\right) \]

5. Applied rewrites 79.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y + x\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]

   2. lower--.f64 38.0

      \[\leadsto b \cdot \color{blue}{\left(a - 0.5\right)} \]

8. Applied rewrites 38.0%

   \[\leadsto \color{blue}{b \cdot \left(a - 0.5\right)} \]
9. Add Preprocessing

Alternative 15: 25.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ b \cdot a \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* b a))

C

double code(double x, double y, double z, double t, double a, double b) {
	return b * a;
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return b * a;
}

Python

def code(x, y, z, t, a, b):
	return b * a

Julia

function code(x, y, z, t, a, b)
	return Float64(b * a)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = b * a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * a), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{b \cdot a} \]

   2. lower-*.f64 28.3

      \[\leadsto \color{blue}{b \cdot a} \]

5. Applied rewrites 28.3%

   \[\leadsto \color{blue}{b \cdot a} \]
6. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + y) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + log(t)))) + ((a - 0.5d0) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 1/2) b)))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))