Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Percentage Accurate: 97.6% → 98.1%

Time: 2.4s

Alternatives: 7

Speedup: 1.1×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -620:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -620.0) t_0 (if (<= x 1.4e-24) (fma y x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -620.0) {
		tmp = t_0;
	} else if (x <= 1.4e-24) {
		tmp = fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -620.0)
		tmp = t_0;
	elseif (x <= 1.4e-24)
		tmp = fma(y, x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -620.0], t$95$0, If[LessEqual[x, 1.4e-24], N[(y * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -620 or 1.4000000000000001e-24 < x

   1. Initial program 95.5%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot \left(y + -1 \cdot z\right) \]

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(y - 1 \cdot z\right) \cdot x \]

      7. *-lft-identity N/A

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

      8. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   if -620 < x < 1.4000000000000001e-24

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 2: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;\left(1 - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-131)
   (fma y x z)
   (if (<= y 8.5e-46) (* (- 1.0 x) z) (fma y x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-131) {
		tmp = fma(y, x, z);
	} else if (y <= 8.5e-46) {
		tmp = (1.0 - x) * z;
	} else {
		tmp = fma(y, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-131)
		tmp = fma(y, x, z);
	elseif (y <= 8.5e-46)
		tmp = Float64(Float64(1.0 - x) * z);
	else
		tmp = fma(y, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-131], N[(y * x + z), $MachinePrecision], If[LessEqual[y, 8.5e-46], N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision], N[(y * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999999e-132 or 8.5000000000000001e-46 < y

   1. Initial program 96.4%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -9.9999999999999999e-132 < y < 8.5000000000000001e-46

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 88.2

         \[\leadsto \left(1 - x\right) \cdot z \]

   5. Applied rewrites 88.2%

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

Alternative 3: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e-9) (* y x) (if (<= x 1.4e-24) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-9) {
		tmp = y * x;
	} else if (x <= 1.4e-24) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	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) :: tmp
    if (x <= (-1.6d-9)) then
        tmp = y * x
    else if (x <= 1.4d-24) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-9) {
		tmp = y * x;
	} else if (x <= 1.4e-24) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e-9:
		tmp = y * x
	elif x <= 1.4e-24:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e-9)
		tmp = Float64(y * x);
	elseif (x <= 1.4e-24)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e-9)
		tmp = y * x;
	elseif (x <= 1.4e-24)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e-9], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.4e-24], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000006e-9 or 1.4000000000000001e-24 < x

   1. Initial program 95.6%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

   if -1.60000000000000006e-9 < x < 1.4000000000000001e-24

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.8%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - x, z, y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- 1.0 x) z (* y x)))

C

double code(double x, double y, double z) {
	return fma((1.0 - x), z, (y * x));
}

Julia

function code(x, y, z)
	return fma(Float64(1.0 - x), z, Float64(y * x))
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 - x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.8

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, z, y \cdot x\right)} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.45e+134) (fma y x z) (* (- x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.45e+134) {
		tmp = fma(y, x, z);
	} else {
		tmp = -x * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.45e+134)
		tmp = fma(y, x, z);
	else
		tmp = Float64(Float64(-x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.45e+134], N[(y * x + z), $MachinePrecision], N[((-x) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.44999999999999998e134

   1. Initial program 98.5%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 2.44999999999999998e134 < x

   1. Initial program 92.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 53.8

         \[\leadsto \left(1 - x\right) \cdot z \]

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot z \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.8

         \[\leadsto \left(-x\right) \cdot z \]

   8. Applied rewrites 53.8%

      \[\leadsto \left(-x\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 76.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 76.1

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

7. Applied rewrites 76.1%

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

Alternative 7: 36.4% accurate, 17.0× 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 97.6%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 36.4%

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

Reproduce

herbie shell --seed 2025091 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))