FastMath dist4

Percentage Accurate: 88.1% → 98.5%

Time: 3.6s

Alternatives: 13

Speedup: 0.5×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1))))
   (if (<= t_0 INFINITY) t_0 (* (- (fma -1.0 d1 d4) d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (fma(-1.0, d1, d4) - d3) * d1;
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(-1.0, d1, d4) - d3) * d1);
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(-1.0 * d1 + d4), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1))

   1. Initial program 0.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 79.3

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 89.7%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \leq \infty:\\ \;\;\;\;d1 \cdot \left(d4 + d2\right) - \mathsf{fma}\left(d3, d1, d1 \cdot d1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)) INFINITY)
   (- (* d1 (+ d4 d2)) (fma d3 d1 (* d1 d1)))
   (* (- (fma -1.0 d1 d4) d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (((((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)) <= ((double) INFINITY)) {
		tmp = (d1 * (d4 + d2)) - fma(d3, d1, (d1 * d1));
	} else {
		tmp = (fma(-1.0, d1, d4) - d3) * d1;
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1)) <= Inf)
		tmp = Float64(Float64(d1 * Float64(d4 + d2)) - fma(d3, d1, Float64(d1 * d1)));
	else
		tmp = Float64(Float64(fma(-1.0, d1, d4) - d3) * d1);
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(d1 * N[(d4 + d2), $MachinePrecision]), $MachinePrecision] - N[(d3 * d1 + N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * d1 + d4), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

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

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      9. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      11. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{d1 \cdot d1} \]

      14. pow2 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

      15. +-commutative N/A

          \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 + d1 \cdot d4\right) + -1 \cdot \left(d1 \cdot d3\right)\right)} - {d1}^{2} \]

      16. associate-*r* N/A

          \[\leadsto \left(\left(d1 \cdot d2 + d1 \cdot d4\right) + \color{blue}{\left(-1 \cdot d1\right) \cdot d3}\right) - {d1}^{2} \]

      17. mul-1-neg N/A

          \[\leadsto \left(\left(d1 \cdot d2 + d1 \cdot d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d3\right) - {d1}^{2} \]

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

          \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 + d1 \cdot d4\right) - d1 \cdot d3\right)} - {d1}^{2} \]

      19. associate--r+ N/A

          \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - \left(d1 \cdot d3 + {d1}^{2}\right)} \]

      20. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - \left(d1 \cdot d3 + {d1}^{2}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d4 + d2\right) - \mathsf{fma}\left(d3, d1, d1 \cdot d1\right)} \]

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1))

   1. Initial program 0.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 79.3

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 89.7%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-d1\right) \cdot d1\\ \mathbf{if}\;d1 \leq -4.3 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d1 \leq 4.6 \cdot 10^{-200}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 1.35 \cdot 10^{+156}:\\ \;\;\;\;\left(d2 + d4\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* (- d1) d1)))
   (if (<= d1 -4.3e+28)
     t_0
     (if (<= d1 4.6e-200)
       (* (- d2 d3) d1)
       (if (<= d1 1.35e+156) (* (+ d2 d4) d1) t_0)))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = -d1 * d1;
	double tmp;
	if (d1 <= -4.3e+28) {
		tmp = t_0;
	} else if (d1 <= 4.6e-200) {
		tmp = (d2 - d3) * d1;
	} else if (d1 <= 1.35e+156) {
		tmp = (d2 + d4) * d1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -d1 * d1
    if (d1 <= (-4.3d+28)) then
        tmp = t_0
    else if (d1 <= 4.6d-200) then
        tmp = (d2 - d3) * d1
    else if (d1 <= 1.35d+156) then
        tmp = (d2 + d4) * d1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = -d1 * d1;
	double tmp;
	if (d1 <= -4.3e+28) {
		tmp = t_0;
	} else if (d1 <= 4.6e-200) {
		tmp = (d2 - d3) * d1;
	} else if (d1 <= 1.35e+156) {
		tmp = (d2 + d4) * d1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = -d1 * d1
	tmp = 0
	if d1 <= -4.3e+28:
		tmp = t_0
	elif d1 <= 4.6e-200:
		tmp = (d2 - d3) * d1
	elif d1 <= 1.35e+156:
		tmp = (d2 + d4) * d1
	else:
		tmp = t_0
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(Float64(-d1) * d1)
	tmp = 0.0
	if (d1 <= -4.3e+28)
		tmp = t_0;
	elseif (d1 <= 4.6e-200)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d1 <= 1.35e+156)
		tmp = Float64(Float64(d2 + d4) * d1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = -d1 * d1;
	tmp = 0.0;
	if (d1 <= -4.3e+28)
		tmp = t_0;
	elseif (d1 <= 4.6e-200)
		tmp = (d2 - d3) * d1;
	elseif (d1 <= 1.35e+156)
		tmp = (d2 + d4) * d1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[((-d1) * d1), $MachinePrecision]}, If[LessEqual[d1, -4.3e+28], t$95$0, If[LessEqual[d1, 4.6e-200], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d1, 1.35e+156], N[(N[(d2 + d4), $MachinePrecision] * d1), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -4.29999999999999975e28 or 1.35e156 < d1

   1. Initial program 71.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 82.6

         \[\leadsto \left(-d1\right) \cdot d1 \]

   5. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if -4.29999999999999975e28 < d1 < 4.60000000000000015e-200

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 94.8

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around inf

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]

   if 4.60000000000000015e-200 < d1 < 1.35e156

   1. Initial program 96.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 86.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d3 around 0

      \[\leadsto \left(d2 + d4\right) \cdot d1 \]
   7. Step-by-step derivation

      1. lower-+.f64 64.4

         \[\leadsto \left(d2 + d4\right) \cdot d1 \]

   8. Applied rewrites 64.4%

      \[\leadsto \left(d2 + d4\right) \cdot d1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 38.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -4.1 \cdot 10^{+107}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d2 \leq -2.7 \cdot 10^{-164}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{elif}\;d2 \leq 2.4 \cdot 10^{-305}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -4.1e+107)
   (* d2 d1)
   (if (<= d2 -2.7e-164)
     (* (- d1) d1)
     (if (<= d2 2.4e-305) (* (- d3) d1) (* d4 d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4.1e+107) {
		tmp = d2 * d1;
	} else if (d2 <= -2.7e-164) {
		tmp = -d1 * d1;
	} else if (d2 <= 2.4e-305) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-4.1d+107)) then
        tmp = d2 * d1
    else if (d2 <= (-2.7d-164)) then
        tmp = -d1 * d1
    else if (d2 <= 2.4d-305) then
        tmp = -d3 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4.1e+107) {
		tmp = d2 * d1;
	} else if (d2 <= -2.7e-164) {
		tmp = -d1 * d1;
	} else if (d2 <= 2.4e-305) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -4.1e+107:
		tmp = d2 * d1
	elif d2 <= -2.7e-164:
		tmp = -d1 * d1
	elif d2 <= 2.4e-305:
		tmp = -d3 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -4.1e+107)
		tmp = Float64(d2 * d1);
	elseif (d2 <= -2.7e-164)
		tmp = Float64(Float64(-d1) * d1);
	elseif (d2 <= 2.4e-305)
		tmp = Float64(Float64(-d3) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -4.1e+107)
		tmp = d2 * d1;
	elseif (d2 <= -2.7e-164)
		tmp = -d1 * d1;
	elseif (d2 <= 2.4e-305)
		tmp = -d3 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -4.1e+107], N[(d2 * d1), $MachinePrecision], If[LessEqual[d2, -2.7e-164], N[((-d1) * d1), $MachinePrecision], If[LessEqual[d2, 2.4e-305], N[((-d3) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d2 < -4.0999999999999999e107

   1. Initial program 81.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 72.6

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -4.0999999999999999e107 < d2 < -2.7000000000000001e-164

   1. Initial program 95.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 44.6

         \[\leadsto \left(-d1\right) \cdot d1 \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if -2.7000000000000001e-164 < d2 < 2.40000000000000019e-305

   1. Initial program 91.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d3 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d3\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d3\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d3\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d3\right)\right) \cdot d1 \]

      7. lower-neg.f64 40.8

         \[\leadsto \left(-d3\right) \cdot d1 \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\left(-d3\right) \cdot d1} \]

   if 2.40000000000000019e-305 < d2

   1. Initial program 86.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d4 \cdot \color{blue}{d1} \]

      2. lift-*.f64 30.2

         \[\leadsto d4 \cdot \color{blue}{d1} \]

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -2.25 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 1.35 \cdot 10^{+156}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, d1, d4\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d1 -2.25e+29)
   (* (- (- d1) d3) d1)
   (if (<= d1 1.35e+156) (* (- (+ d4 d2) d3) d1) (* (fma -1.0 d1 d4) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -2.25e+29) {
		tmp = (-d1 - d3) * d1;
	} else if (d1 <= 1.35e+156) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = fma(-1.0, d1, d4) * d1;
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d1 <= -2.25e+29)
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	elseif (d1 <= 1.35e+156)
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	else
		tmp = Float64(fma(-1.0, d1, d4) * d1);
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d1, -2.25e+29], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d1, 1.35e+156], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(-1.0 * d1 + d4), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -2.2500000000000001e29

   1. Initial program 79.1%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 87.8

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 87.8

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 87.8%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]

   9. Taylor expanded in d1 around inf

      \[\leadsto \left(-1 \cdot d1 - d3\right) \cdot d1 \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(d1\right)\right) - d3\right) \cdot d1 \]

       2. lower-neg.f64 85.4

          \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   11. Applied rewrites 85.4%

       \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   if -2.2500000000000001e29 < d1 < 1.35e156

   1. Initial program 98.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 91.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   if 1.35e156 < d1

   1. Initial program 63.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 93.5

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 97.8

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]

   9. Taylor expanded in d3 around 0

      \[\leadsto -1 \cdot {d1}^{2} + \color{blue}{d1 \cdot d4} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto d1 \cdot d4 + -1 \cdot \color{blue}{{d1}^{2}} \]

       2. mul-1-neg N/A

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

       3. pow2 N/A

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

       5. mul-1-neg N/A

          \[\leadsto d1 \cdot d4 + \left(-1 \cdot d1\right) \cdot d1 \]

       6. *-commutative N/A

          \[\leadsto d1 \cdot d4 + d1 \cdot \left(-1 \cdot \color{blue}{d1}\right) \]

       7. distribute-lft-in N/A

          \[\leadsto d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(d4 + -1 \cdot d1\right) \cdot d1 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(d4 + -1 \cdot d1\right) \cdot d1 \]

       10. +-commutative N/A

           \[\leadsto \left(-1 \cdot d1 + d4\right) \cdot d1 \]

       11. lift-fma.f64 95.7

           \[\leadsto \mathsf{fma}\left(-1, d1, d4\right) \cdot d1 \]

   11. Applied rewrites 95.7%

       \[\leadsto \mathsf{fma}\left(-1, d1, d4\right) \cdot \color{blue}{d1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.05 \cdot 10^{+116}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, d1, d4\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.05e+116)
   (* (- d2 d3) d1)
   (if (<= d2 1.9e-304) (* (- (- d1) d3) d1) (* (fma -1.0 d1 d4) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.05e+116) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= 1.9e-304) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = fma(-1.0, d1, d4) * d1;
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.05e+116)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= 1.9e-304)
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	else
		tmp = Float64(fma(-1.0, d1, d4) * d1);
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.05e+116], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, 1.9e-304], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(-1.0 * d1 + d4), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -1.0500000000000001e116

   1. Initial program 81.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 93.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around inf

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 90.0%

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]

   if -1.0500000000000001e116 < d2 < 1.8999999999999998e-304

   1. Initial program 94.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 93.5

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 93.5

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 93.5%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]

   9. Taylor expanded in d1 around inf

      \[\leadsto \left(-1 \cdot d1 - d3\right) \cdot d1 \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(d1\right)\right) - d3\right) \cdot d1 \]

       2. lower-neg.f64 71.7

          \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   11. Applied rewrites 71.7%

       \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   if 1.8999999999999998e-304 < d2

   1. Initial program 86.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 79.5

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 80.9

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 80.9%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]

   9. Taylor expanded in d3 around 0

      \[\leadsto -1 \cdot {d1}^{2} + \color{blue}{d1 \cdot d4} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto d1 \cdot d4 + -1 \cdot \color{blue}{{d1}^{2}} \]

       2. mul-1-neg N/A

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

       3. pow2 N/A

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto d1 \cdot d4 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

       5. mul-1-neg N/A

          \[\leadsto d1 \cdot d4 + \left(-1 \cdot d1\right) \cdot d1 \]

       6. *-commutative N/A

          \[\leadsto d1 \cdot d4 + d1 \cdot \left(-1 \cdot \color{blue}{d1}\right) \]

       7. distribute-lft-in N/A

          \[\leadsto d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(d4 + -1 \cdot d1\right) \cdot d1 \]

       9. lower-*.f64 N/A

          \[\leadsto \left(d4 + -1 \cdot d1\right) \cdot d1 \]

       10. +-commutative N/A

           \[\leadsto \left(-1 \cdot d1 + d4\right) \cdot d1 \]

       11. lift-fma.f64 59.3

           \[\leadsto \mathsf{fma}\left(-1, d1, d4\right) \cdot d1 \]

   11. Applied rewrites 59.3%

       \[\leadsto \mathsf{fma}\left(-1, d1, d4\right) \cdot \color{blue}{d1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.05 \cdot 10^{+116}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq 2.15 \cdot 10^{-174}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.05e+116)
   (* (- d2 d3) d1)
   (if (<= d2 2.15e-174) (* (- (- d1) d3) d1) (* (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.05e+116) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= 2.15e-174) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-1.05d+116)) then
        tmp = (d2 - d3) * d1
    else if (d2 <= 2.15d-174) then
        tmp = (-d1 - d3) * d1
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.05e+116) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= 2.15e-174) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -1.05e+116:
		tmp = (d2 - d3) * d1
	elif d2 <= 2.15e-174:
		tmp = (-d1 - d3) * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.05e+116)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= 2.15e-174)
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -1.05e+116)
		tmp = (d2 - d3) * d1;
	elseif (d2 <= 2.15e-174)
		tmp = (-d1 - d3) * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.05e+116], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, 2.15e-174], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -1.0500000000000001e116

   1. Initial program 81.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 93.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around inf

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 90.0%

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]

   if -1.0500000000000001e116 < d2 < 2.1500000000000002e-174

   1. Initial program 91.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 94.6

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 95.4

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 95.4%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]

   9. Taylor expanded in d1 around inf

      \[\leadsto \left(-1 \cdot d1 - d3\right) \cdot d1 \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(d1\right)\right) - d3\right) \cdot d1 \]

       2. lower-neg.f64 74.3

          \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   11. Applied rewrites 74.3%

       \[\leadsto \left(\left(-d1\right) - d3\right) \cdot d1 \]

   if 2.1500000000000002e-174 < d2

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 78.2

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 51.5%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -2.25 \cdot 10^{+29} \lor \neg \left(d1 \leq 1.35 \cdot 10^{+156}\right):\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d2 + d4\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (or (<= d1 -2.25e+29) (not (<= d1 1.35e+156)))
   (* (- d1) d1)
   (* (+ d2 d4) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -2.25e+29) || !(d1 <= 1.35e+156)) {
		tmp = -d1 * d1;
	} else {
		tmp = (d2 + d4) * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if ((d1 <= (-2.25d+29)) .or. (.not. (d1 <= 1.35d+156))) then
        tmp = -d1 * d1
    else
        tmp = (d2 + d4) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -2.25e+29) || !(d1 <= 1.35e+156)) {
		tmp = -d1 * d1;
	} else {
		tmp = (d2 + d4) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if (d1 <= -2.25e+29) or not (d1 <= 1.35e+156):
		tmp = -d1 * d1
	else:
		tmp = (d2 + d4) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if ((d1 <= -2.25e+29) || !(d1 <= 1.35e+156))
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(Float64(d2 + d4) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if ((d1 <= -2.25e+29) || ~((d1 <= 1.35e+156)))
		tmp = -d1 * d1;
	else
		tmp = (d2 + d4) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[Or[LessEqual[d1, -2.25e+29], N[Not[LessEqual[d1, 1.35e+156]], $MachinePrecision]], N[((-d1) * d1), $MachinePrecision], N[(N[(d2 + d4), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d1 < -2.2500000000000001e29 or 1.35e156 < d1

   1. Initial program 71.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 83.4

         \[\leadsto \left(-d1\right) \cdot d1 \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if -2.2500000000000001e29 < d1 < 1.35e156

   1. Initial program 98.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 91.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d3 around 0

      \[\leadsto \left(d2 + d4\right) \cdot d1 \]
   7. Step-by-step derivation

      1. lower-+.f64 60.6

         \[\leadsto \left(d2 + d4\right) \cdot d1 \]

   8. Applied rewrites 60.6%

      \[\leadsto \left(d2 + d4\right) \cdot d1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d1 \leq -2.25 \cdot 10^{+29} \lor \neg \left(d1 \leq 1.35 \cdot 10^{+156}\right):\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d2 + d4\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -6.2 \cdot 10^{-117}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.95e-7)
   (* (- d2 d3) d1)
   (if (<= d2 -6.2e-117) (* (- d1) d1) (* (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.95e-7) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -6.2e-117) {
		tmp = -d1 * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-1.95d-7)) then
        tmp = (d2 - d3) * d1
    else if (d2 <= (-6.2d-117)) then
        tmp = -d1 * d1
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.95e-7) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -6.2e-117) {
		tmp = -d1 * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -1.95e-7:
		tmp = (d2 - d3) * d1
	elif d2 <= -6.2e-117:
		tmp = -d1 * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.95e-7)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= -6.2e-117)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -1.95e-7)
		tmp = (d2 - d3) * d1;
	elseif (d2 <= -6.2e-117)
		tmp = -d1 * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.95e-7], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, -6.2e-117], N[((-d1) * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -1.95000000000000012e-7

   1. Initial program 86.5%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 90.4

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around inf

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 77.2%

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]

   if -1.95000000000000012e-7 < d2 < -6.20000000000000022e-117

   1. Initial program 93.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 57.3

         \[\leadsto \left(-d1\right) \cdot d1 \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if -6.20000000000000022e-117 < d2

   1. Initial program 88.5%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 72.0

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   6. Taylor expanded in d2 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.05 \cdot 10^{+116}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.05e+116)
   (* (- (+ d4 d2) d3) d1)
   (* (- (fma -1.0 d1 d4) d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.05e+116) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = (fma(-1.0, d1, d4) - d3) * d1;
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.05e+116)
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	else
		tmp = Float64(Float64(fma(-1.0, d1, d4) - d3) * d1);
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.05e+116], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(N[(-1.0 * d1 + d4), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.0500000000000001e116

   1. Initial program 81.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 93.5

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   if -1.0500000000000001e116 < d2

   1. Initial program 89.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. pow2 N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - d1 \cdot \color{blue}{d1} \]

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

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right) \cdot d1} \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)} \cdot d1 \]

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

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left(d1 \cdot d1\right)\right) \]

      6. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) + \left(\mathsf{neg}\left({d1}^{2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, \color{blue}{d4 - d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - \color{blue}{d3}, \mathsf{neg}\left({d1}^{2}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \mathsf{neg}\left(d1 \cdot d1\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-1 \cdot d1\right) \cdot d1\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(\mathsf{neg}\left(d1\right)\right) \cdot d1\right) \]

      14. lower-neg.f64 84.9

          \[\leadsto \mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, d4 - d3, \left(-d1\right) \cdot d1\right)} \]

   6. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 + -1 \cdot d1\right) - d3\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d4 + -1 \cdot d1\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d1 + d4\right) - d3\right) \cdot d1 \]

      5. lower-fma.f64 85.8

         \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot d1 \]

   8. Applied rewrites 85.8%

      \[\leadsto \left(\mathsf{fma}\left(-1, d1, d4\right) - d3\right) \cdot \color{blue}{d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -4.1 \cdot 10^{+107}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d2 \leq 2 \cdot 10^{-165}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -4.1e+107) (* d2 d1) (if (<= d2 2e-165) (* (- d1) d1) (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4.1e+107) {
		tmp = d2 * d1;
	} else if (d2 <= 2e-165) {
		tmp = -d1 * d1;
	} else {
		tmp = d4 * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-4.1d+107)) then
        tmp = d2 * d1
    else if (d2 <= 2d-165) then
        tmp = -d1 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4.1e+107) {
		tmp = d2 * d1;
	} else if (d2 <= 2e-165) {
		tmp = -d1 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -4.1e+107:
		tmp = d2 * d1
	elif d2 <= 2e-165:
		tmp = -d1 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -4.1e+107)
		tmp = Float64(d2 * d1);
	elseif (d2 <= 2e-165)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -4.1e+107)
		tmp = d2 * d1;
	elseif (d2 <= 2e-165)
		tmp = -d1 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -4.1e+107], N[(d2 * d1), $MachinePrecision], If[LessEqual[d2, 2e-165], N[((-d1) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -4.0999999999999999e107

   1. Initial program 81.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 72.6

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -4.0999999999999999e107 < d2 < 2e-165

   1. Initial program 91.8%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 47.8

         \[\leadsto \left(-d1\right) \cdot d1 \]

   5. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if 2e-165 < d2

   1. Initial program 87.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d4 \cdot \color{blue}{d1} \]

      2. lift-*.f64 31.5

         \[\leadsto d4 \cdot \color{blue}{d1} \]

   5. Applied rewrites 31.5%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -0.00192:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -0.00192) (* d2 d1) (* d4 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -0.00192) {
		tmp = d2 * d1;
	} else {
		tmp = d4 * d1;
	}
	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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-0.00192d0)) then
        tmp = d2 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -0.00192) {
		tmp = d2 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -0.00192:
		tmp = d2 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -0.00192)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -0.00192)
		tmp = d2 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -0.00192], N[(d2 * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -0.00192000000000000005

   1. Initial program 86.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 58.5

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -0.00192000000000000005 < d2

   1. Initial program 89.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d4 \cdot \color{blue}{d1} \]

      2. lift-*.f64 30.7

         \[\leadsto d4 \cdot \color{blue}{d1} \]

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 30.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ d2 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d2 d1))

C

double code(double d1, double d2, double d3, double d4) {
	return d2 * d1;
}

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(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d2 * d1
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d2 * d1;
}

Python

def code(d1, d2, d3, d4):
	return d2 * d1

Julia

function code(d1, d2, d3, d4)
	return Float64(d2 * d1)
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d2 * d1;
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d2 * d1), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
2. Add Preprocessing

3. Taylor expanded in d2 around inf

   \[\leadsto \color{blue}{d1 \cdot d2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d2 \cdot \color{blue}{d1} \]

   2. lower-*.f64 26.7

      \[\leadsto d2 \cdot \color{blue}{d1} \]

5. Applied rewrites 26.7%

   \[\leadsto \color{blue}{d2 \cdot d1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

  :alt
  (! :herbie-platform default (* d1 (- (+ (- d2 d3) d4) d1)))

  (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))