Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 7

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, \mathsf{fma}\left(0.5, x, z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y x (fma 0.5 x z)))

C

double code(double x, double y, double z) {
	return fma(y, x, fma(0.5, x, z));
}

Julia

function code(x, y, z)
	return fma(y, x, fma(0.5, x, z))
end

Wolfram

code[x_, y_, z_] := N[(y * x + N[(0.5 * x + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot x + \frac{x}{2}\right)} + z \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{y \cdot x + \left(\frac{x}{2} + z\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot x} + \left(\frac{x}{2} + z\right) \]

   6. +-commutative N/A

      \[\leadsto y \cdot x + \color{blue}{\left(z + \frac{x}{2}\right)} \]

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. add-flip-rev N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{x}{2} - \left(\mathsf{neg}\left(z\right)\right)}\right) \]

   10. sub-flip N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{x}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\frac{x}{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right) \]

   12. mult-flip N/A

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

   13. *-commutative N/A

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

   14. remove-double-neg N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(\color{blue}{0.5}, x, z\right)\right) \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(0.5, x, z\right)\right)} \]
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(y - -0.5), $MachinePrecision] * x + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot x + \frac{x}{2}\right)} + z \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{y \cdot x} + \frac{x}{2}\right) + z \]

   5. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot y} + \frac{x}{2}\right) + z \]

   6. lift-/.f64 N/A

      \[\leadsto \left(x \cdot y + \color{blue}{\frac{x}{2}}\right) + z \]

   7. mult-flip N/A

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

   8. distribute-lft-out N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. add-flip N/A

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

   12. distribute-frac-neg2 N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 100.0

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - -0.5, x, z\right)} \]
4. Add Preprocessing

Alternative 3: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + y \cdot x\\ t_1 := \left(y - -0.5\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* y x))) (t_1 (* (- y -0.5) x)))
   (if (<= t_0 -5e+195) t_1 (if (<= t_0 5e+153) (fma 0.5 x z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double t_1 = (y - -0.5) * x;
	double tmp;
	if (t_0 <= -5e+195) {
		tmp = t_1;
	} else if (t_0 <= 5e+153) {
		tmp = fma(0.5, x, z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(y * x))
	t_1 = Float64(Float64(y - -0.5) * x)
	tmp = 0.0
	if (t_0 <= -5e+195)
		tmp = t_1;
	elseif (t_0 <= 5e+153)
		tmp = fma(0.5, x, z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y - -0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+195], t$95$1, If[LessEqual[t$95$0, 5e+153], N[(0.5 * x + z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -4.9999999999999998e195 or 5.00000000000000018e153 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 60.1

         \[\leadsto x \cdot \left(0.5 + \color{blue}{y}\right) \]

   7. Applied rewrites 60.1%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.1

         \[\leadsto \left(0.5 + y\right) \cdot \color{blue}{x} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + y\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \left(y + \frac{1}{2}\right) \cdot x \]

      6. metadata-eval N/A

         \[\leadsto \left(y + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right) \cdot x \]

      7. sub-flip N/A

         \[\leadsto \left(y - \frac{-1}{2}\right) \cdot x \]

      8. lower--.f64 60.1

         \[\leadsto \left(y - -0.5\right) \cdot x \]

   9. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\left(y - -0.5\right) \cdot x} \]

   if -4.9999999999999998e195 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x + \frac{x}{2}\right)} + z \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{y \cdot x} + \frac{x}{2}\right) + z \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot y} + \frac{x}{2}\right) + z \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\frac{x}{2}}\right) + z \]

      7. mult-flip N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. add-flip N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 65.2%

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

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+166}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.26e+166) (* x y) (if (<= y 1.1e+25) (fma 0.5 x z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.26e+166) {
		tmp = x * y;
	} else if (y <= 1.1e+25) {
		tmp = fma(0.5, x, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.26e+166)
		tmp = Float64(x * y);
	elseif (y <= 1.1e+25)
		tmp = fma(0.5, x, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.26e+166], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.1e+25], N[(0.5 * x + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.26e166 or 1.1e25 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 36.5

         \[\leadsto x \cdot \color{blue}{y} \]

   7. Applied rewrites 36.5%

      \[\leadsto \color{blue}{x \cdot y} \]

   if -1.26e166 < y < 1.1e25

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right) + z} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{2} + y \cdot x\right)} + z \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x + \frac{x}{2}\right)} + z \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{y \cdot x} + \frac{x}{2}\right) + z \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot y} + \frac{x}{2}\right) + z \]

      6. lift-/.f64 N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\frac{x}{2}}\right) + z \]

      7. mult-flip N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. add-flip N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 65.2%

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

Alternative 5: 60.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + y \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+296}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+92}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* y x))))
   (if (<= t_0 -5e+296)
     (* x y)
     (if (<= t_0 -5e+92)
       (* x 0.5)
       (if (<= t_0 5e+153) z (if (<= t_0 2e+263) (* x 0.5) (* x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double tmp;
	if (t_0 <= -5e+296) {
		tmp = x * y;
	} else if (t_0 <= -5e+92) {
		tmp = x * 0.5;
	} else if (t_0 <= 5e+153) {
		tmp = z;
	} else if (t_0 <= 2e+263) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / 2.0d0) + (y * x)
    if (t_0 <= (-5d+296)) then
        tmp = x * y
    else if (t_0 <= (-5d+92)) then
        tmp = x * 0.5d0
    else if (t_0 <= 5d+153) then
        tmp = z
    else if (t_0 <= 2d+263) then
        tmp = x * 0.5d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double tmp;
	if (t_0 <= -5e+296) {
		tmp = x * y;
	} else if (t_0 <= -5e+92) {
		tmp = x * 0.5;
	} else if (t_0 <= 5e+153) {
		tmp = z;
	} else if (t_0 <= 2e+263) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / 2.0) + (y * x)
	tmp = 0
	if t_0 <= -5e+296:
		tmp = x * y
	elif t_0 <= -5e+92:
		tmp = x * 0.5
	elif t_0 <= 5e+153:
		tmp = z
	elif t_0 <= 2e+263:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(y * x))
	tmp = 0.0
	if (t_0 <= -5e+296)
		tmp = Float64(x * y);
	elseif (t_0 <= -5e+92)
		tmp = Float64(x * 0.5);
	elseif (t_0 <= 5e+153)
		tmp = z;
	elseif (t_0 <= 2e+263)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / 2.0) + (y * x);
	tmp = 0.0;
	if (t_0 <= -5e+296)
		tmp = x * y;
	elseif (t_0 <= -5e+92)
		tmp = x * 0.5;
	elseif (t_0 <= 5e+153)
		tmp = z;
	elseif (t_0 <= 2e+263)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+296], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, -5e+92], N[(x * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+153], z, If[LessEqual[t$95$0, 2e+263], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -5.0000000000000001e296 or 2.00000000000000003e263 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 36.5

         \[\leadsto x \cdot \color{blue}{y} \]

   7. Applied rewrites 36.5%

      \[\leadsto \color{blue}{x \cdot y} \]

   if -5.0000000000000001e296 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -5.00000000000000022e92 or 5.00000000000000018e153 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 2.00000000000000003e263

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 60.1

         \[\leadsto x \cdot \left(0.5 + \color{blue}{y}\right) \]

   7. Applied rewrites 60.1%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 25.8%

       \[\leadsto x \cdot 0.5 \]

   if -5.00000000000000022e92 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

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

Alternative 6: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + y \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+157}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* y x))))
   (if (<= t_0 -5e+203) (* x y) (if (<= t_0 4e+157) z (* x y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double tmp;
	if (t_0 <= -5e+203) {
		tmp = x * y;
	} else if (t_0 <= 4e+157) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / 2.0d0) + (y * x)
    if (t_0 <= (-5d+203)) then
        tmp = x * y
    else if (t_0 <= 4d+157) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double tmp;
	if (t_0 <= -5e+203) {
		tmp = x * y;
	} else if (t_0 <= 4e+157) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / 2.0) + (y * x)
	tmp = 0
	if t_0 <= -5e+203:
		tmp = x * y
	elif t_0 <= 4e+157:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(y * x))
	tmp = 0.0
	if (t_0 <= -5e+203)
		tmp = Float64(x * y);
	elseif (t_0 <= 4e+157)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / 2.0) + (y * x);
	tmp = 0.0;
	if (t_0 <= -5e+203)
		tmp = x * y;
	elseif (t_0 <= 4e+157)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+203], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 4e+157], z, N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -4.99999999999999994e203 or 3.99999999999999993e157 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 36.5

         \[\leadsto x \cdot \color{blue}{y} \]

   7. Applied rewrites 36.5%

      \[\leadsto \color{blue}{x \cdot y} \]

   if -4.99999999999999994e203 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 3.99999999999999993e157

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 41.4%

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

Alternative 7: 41.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	return z;
}

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z} \]
3. Step-by-step derivation

4. Applied rewrites 41.4%

   \[\leadsto \color{blue}{z} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))