Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (y + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (y + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (y + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (y + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y 1.0)))

C

double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (y + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (y + 1.0);
}

Python

def code(x, y):
	return (x + y) / (y + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (y + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{y + 1} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -5000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1 - y, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -5000000000.0)
     t_1
     (if (<= t_0 1e-8)
       (fma (- 1.0 y) y x)
       (if (<= t_0 2.0) (/ y (- y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -5000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-8) {
		tmp = fma((1.0 - y), y, x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -5000000000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-8)
		tmp = fma(Float64(1.0 - y), y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000000000.0], t$95$1, If[LessEqual[t$95$0, 1e-8], N[(N[(1.0 - y), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -5e9 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + y}} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{1}{y}} + y} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot \frac{1}{y} + \color{blue}{y \cdot 1}} \]

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot 1 + y \cdot \frac{1}{y}}} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot \frac{1}{y}} \]

      8. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{y + \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{x}{y + \color{blue}{1 \cdot 1}} \]

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

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

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1} \cdot 1} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1}} \]

      13. lower--.f64 99.6

          \[\leadsto \frac{x}{\color{blue}{y - -1}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{x}{y - -1}} \]

   if -5e9 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1e-8

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(y - 1\right), y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if 1e-8 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{1 + y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{1 + y}} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \frac{y}{\color{blue}{y \cdot \frac{1}{y}} + y} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y}{y \cdot \frac{1}{y} + \color{blue}{y \cdot 1}} \]

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{y \cdot 1 + y \cdot \frac{1}{y}}} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{y} + y \cdot \frac{1}{y}} \]

      8. rgt-mult-inverse N/A

         \[\leadsto \frac{y}{y + \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{y}{y + \color{blue}{1 \cdot 1}} \]

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

          \[\leadsto \frac{y}{\color{blue}{y - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{y}{y - \color{blue}{-1} \cdot 1} \]

      12. metadata-eval N/A

          \[\leadsto \frac{y}{y - \color{blue}{-1}} \]

      13. lower--.f64 99.7

          \[\leadsto \frac{y}{\color{blue}{y - -1}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{y}{y - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-5} \lor \neg \left(y \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - y, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e-5) (not (<= y 4.1e-16)))
   (/ x (- y -1.0))
   (fma (- 1.0 y) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-5) || !(y <= 4.1e-16)) {
		tmp = x / (y - -1.0);
	} else {
		tmp = fma((1.0 - y), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e-5) || !(y <= 4.1e-16))
		tmp = Float64(x / Float64(y - -1.0));
	else
		tmp = fma(Float64(1.0 - y), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e-5], N[Not[LessEqual[y, 4.1e-16]], $MachinePrecision]], N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999943e-5 or 4.10000000000000006e-16 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + y}} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{1}{y}} + y} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot \frac{1}{y} + \color{blue}{y \cdot 1}} \]

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot 1 + y \cdot \frac{1}{y}}} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot \frac{1}{y}} \]

      8. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{y + \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{x}{y + \color{blue}{1 \cdot 1}} \]

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

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

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1} \cdot 1} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1}} \]

      13. lower--.f64 30.6

          \[\leadsto \frac{x}{\color{blue}{y - -1}} \]

   5. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\frac{x}{y - -1}} \]

   if -6.49999999999999943e-5 < y < 4.10000000000000006e-16

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(y - 1\right), y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-5} \lor \neg \left(y \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - y, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 160\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 160.0))) (/ x y) (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 160.0)) {
		tmp = x / y;
	} else {
		tmp = fma((1.0 - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 160.0))
		tmp = Float64(x / y);
	else
		tmp = fma(Float64(1.0 - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 160.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 160 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + y}} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{1}{y}} + y} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot \frac{1}{y} + \color{blue}{y \cdot 1}} \]

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot 1 + y \cdot \frac{1}{y}}} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot \frac{1}{y}} \]

      8. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{y + \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{x}{y + \color{blue}{1 \cdot 1}} \]

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

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

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1} \cdot 1} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{y - \color{blue}{-1}} \]

      13. lower--.f64 26.9

          \[\leadsto \frac{x}{\color{blue}{y - -1}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\frac{x}{y - -1}} \]

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 25.6%

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

   if -1 < y < 160

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 160\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma 1.0 y x))

C

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

Julia

function code(x, y)
	return fma(1.0, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-out N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 51.8%

   \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))