Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x - y}{z - y} \]
4. Add Preprocessing

Alternative 2: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x - y}{z}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 17500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ (- x y) z)))
   (if (<= y -7.5e-22)
     t_0
     (if (<= y 9.8e-214)
       t_1
       (if (<= y 1.25e-99)
         (/ x (- z y))
         (if (<= y 17500000000.0)
           t_1
           (if (<= y 5.8e+156) t_0 (/ y (- y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -7.5e-22) {
		tmp = t_0;
	} else if (y <= 9.8e-214) {
		tmp = t_1;
	} else if (y <= 1.25e-99) {
		tmp = x / (z - y);
	} else if (y <= 17500000000.0) {
		tmp = t_1;
	} else if (y <= 5.8e+156) {
		tmp = t_0;
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = (x - y) / z
    if (y <= (-7.5d-22)) then
        tmp = t_0
    else if (y <= 9.8d-214) then
        tmp = t_1
    else if (y <= 1.25d-99) then
        tmp = x / (z - y)
    else if (y <= 17500000000.0d0) then
        tmp = t_1
    else if (y <= 5.8d+156) then
        tmp = t_0
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -7.5e-22) {
		tmp = t_0;
	} else if (y <= 9.8e-214) {
		tmp = t_1;
	} else if (y <= 1.25e-99) {
		tmp = x / (z - y);
	} else if (y <= 17500000000.0) {
		tmp = t_1;
	} else if (y <= 5.8e+156) {
		tmp = t_0;
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = (x - y) / z
	tmp = 0
	if y <= -7.5e-22:
		tmp = t_0
	elif y <= 9.8e-214:
		tmp = t_1
	elif y <= 1.25e-99:
		tmp = x / (z - y)
	elif y <= 17500000000.0:
		tmp = t_1
	elif y <= 5.8e+156:
		tmp = t_0
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (y <= -7.5e-22)
		tmp = t_0;
	elseif (y <= 9.8e-214)
		tmp = t_1;
	elseif (y <= 1.25e-99)
		tmp = Float64(x / Float64(z - y));
	elseif (y <= 17500000000.0)
		tmp = t_1;
	elseif (y <= 5.8e+156)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	t_1 = (x - y) / z;
	tmp = 0.0;
	if (y <= -7.5e-22)
		tmp = t_0;
	elseif (y <= 9.8e-214)
		tmp = t_1;
	elseif (y <= 1.25e-99)
		tmp = x / (z - y);
	elseif (y <= 17500000000.0)
		tmp = t_1;
	elseif (y <= 5.8e+156)
		tmp = t_0;
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -7.5e-22], t$95$0, If[LessEqual[y, 9.8e-214], t$95$1, If[LessEqual[y, 1.25e-99], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 17500000000.0], t$95$1, If[LessEqual[y, 5.8e+156], t$95$0, N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.49999999999999978e-22 or 1.75e10 < y < 5.80000000000000021e156

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 83.1%

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

      1. div-sub 83.1%

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

      2. sub-neg 83.1%

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

      3. *-inverses 83.1%

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

      4. metadata-eval 83.1%

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

      5. distribute-lft-in 83.1%

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

      6. metadata-eval 83.1%

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

      7. +-commutative 83.1%

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

      8. mul-1-neg 83.1%

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

      9. unsub-neg 83.1%

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

   5. Simplified 83.1%

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

   if -7.49999999999999978e-22 < y < 9.79999999999999935e-214 or 1.24999999999999992e-99 < y < 1.75e10

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 82.6%

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

   if 9.79999999999999935e-214 < y < 1.24999999999999992e-99

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.7%

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

   if 5.80000000000000021e156 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 89.2%

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

      1. neg-mul-1 89.2%

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

      2. distribute-neg-frac 89.2%

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

   5. Simplified 89.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 17500000000:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+156}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -4.6e-25)
     t_0
     (if (<= y 2.15e-141)
       (/ x z)
       (if (<= y 8e-105) (/ (- x) y) (if (<= y 2.05e-25) (/ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -4.6e-25) {
		tmp = t_0;
	} else if (y <= 2.15e-141) {
		tmp = x / z;
	} else if (y <= 8e-105) {
		tmp = -x / y;
	} else if (y <= 2.05e-25) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-4.6d-25)) then
        tmp = t_0
    else if (y <= 2.15d-141) then
        tmp = x / z
    else if (y <= 8d-105) then
        tmp = -x / y
    else if (y <= 2.05d-25) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -4.6e-25) {
		tmp = t_0;
	} else if (y <= 2.15e-141) {
		tmp = x / z;
	} else if (y <= 8e-105) {
		tmp = -x / y;
	} else if (y <= 2.05e-25) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -4.6e-25:
		tmp = t_0
	elif y <= 2.15e-141:
		tmp = x / z
	elif y <= 8e-105:
		tmp = -x / y
	elif y <= 2.05e-25:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -4.6e-25)
		tmp = t_0;
	elseif (y <= 2.15e-141)
		tmp = Float64(x / z);
	elseif (y <= 8e-105)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 2.05e-25)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -4.6e-25)
		tmp = t_0;
	elseif (y <= 2.15e-141)
		tmp = x / z;
	elseif (y <= 8e-105)
		tmp = -x / y;
	elseif (y <= 2.05e-25)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e-25], t$95$0, If[LessEqual[y, 2.15e-141], N[(x / z), $MachinePrecision], If[LessEqual[y, 8e-105], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 2.05e-25], N[(x / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999998e-25 or 2.04999999999999994e-25 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 80.5%

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

      1. div-sub 80.5%

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

      2. sub-neg 80.5%

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

      3. *-inverses 80.5%

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

      4. metadata-eval 80.5%

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

      5. distribute-lft-in 80.5%

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

      6. metadata-eval 80.5%

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

      7. +-commutative 80.5%

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

      8. mul-1-neg 80.5%

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

      9. unsub-neg 80.5%

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

   5. Simplified 80.5%

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

   if -4.5999999999999998e-25 < y < 2.14999999999999987e-141 or 7.99999999999999972e-105 < y < 2.04999999999999994e-25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 67.8%

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

   if 2.14999999999999987e-141 < y < 7.99999999999999972e-105

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.9%

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

   4. Taylor expanded in z around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. distribute-frac-neg2 71.4%

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

   6. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{x}{-y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x - y}{z}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 98000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ (- x y) z)))
   (if (<= y -1.45e-22)
     t_0
     (if (<= y 5.5e-214)
       t_1
       (if (<= y 2.9e-100) (/ x (- z y)) (if (<= y 98000000.0) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -1.45e-22) {
		tmp = t_0;
	} else if (y <= 5.5e-214) {
		tmp = t_1;
	} else if (y <= 2.9e-100) {
		tmp = x / (z - y);
	} else if (y <= 98000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = (x - y) / z
    if (y <= (-1.45d-22)) then
        tmp = t_0
    else if (y <= 5.5d-214) then
        tmp = t_1
    else if (y <= 2.9d-100) then
        tmp = x / (z - y)
    else if (y <= 98000000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -1.45e-22) {
		tmp = t_0;
	} else if (y <= 5.5e-214) {
		tmp = t_1;
	} else if (y <= 2.9e-100) {
		tmp = x / (z - y);
	} else if (y <= 98000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = (x - y) / z
	tmp = 0
	if y <= -1.45e-22:
		tmp = t_0
	elif y <= 5.5e-214:
		tmp = t_1
	elif y <= 2.9e-100:
		tmp = x / (z - y)
	elif y <= 98000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (y <= -1.45e-22)
		tmp = t_0;
	elseif (y <= 5.5e-214)
		tmp = t_1;
	elseif (y <= 2.9e-100)
		tmp = Float64(x / Float64(z - y));
	elseif (y <= 98000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	t_1 = (x - y) / z;
	tmp = 0.0;
	if (y <= -1.45e-22)
		tmp = t_0;
	elseif (y <= 5.5e-214)
		tmp = t_1;
	elseif (y <= 2.9e-100)
		tmp = x / (z - y);
	elseif (y <= 98000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.45e-22], t$95$0, If[LessEqual[y, 5.5e-214], t$95$1, If[LessEqual[y, 2.9e-100], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 98000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4500000000000001e-22 or 9.8e7 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 83.0%

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

      1. div-sub 83.0%

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

      2. sub-neg 83.0%

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

      3. *-inverses 83.0%

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

      4. metadata-eval 83.0%

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

      5. distribute-lft-in 83.0%

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

      6. metadata-eval 83.0%

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

      7. +-commutative 83.0%

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

      8. mul-1-neg 83.0%

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

      9. unsub-neg 83.0%

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

   5. Simplified 83.0%

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

   if -1.4500000000000001e-22 < y < 5.50000000000000024e-214 or 2.89999999999999975e-100 < y < 9.8e7

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 82.6%

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

   if 5.50000000000000024e-214 < y < 2.89999999999999975e-100

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-22}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 98000000:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-104}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e-9)
   1.0
   (if (<= y 2.15e-141)
     (/ x z)
     (if (<= y 1.7e-104) (/ (- x) y) (if (<= y 3.7e-25) (/ x z) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e-9) {
		tmp = 1.0;
	} else if (y <= 2.15e-141) {
		tmp = x / z;
	} else if (y <= 1.7e-104) {
		tmp = -x / y;
	} else if (y <= 3.7e-25) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.6d-9)) then
        tmp = 1.0d0
    else if (y <= 2.15d-141) then
        tmp = x / z
    else if (y <= 1.7d-104) then
        tmp = -x / y
    else if (y <= 3.7d-25) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e-9) {
		tmp = 1.0;
	} else if (y <= 2.15e-141) {
		tmp = x / z;
	} else if (y <= 1.7e-104) {
		tmp = -x / y;
	} else if (y <= 3.7e-25) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6e-9:
		tmp = 1.0
	elif y <= 2.15e-141:
		tmp = x / z
	elif y <= 1.7e-104:
		tmp = -x / y
	elif y <= 3.7e-25:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6e-9)
		tmp = 1.0;
	elseif (y <= 2.15e-141)
		tmp = Float64(x / z);
	elseif (y <= 1.7e-104)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 3.7e-25)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6e-9)
		tmp = 1.0;
	elseif (y <= 2.15e-141)
		tmp = x / z;
	elseif (y <= 1.7e-104)
		tmp = -x / y;
	elseif (y <= 3.7e-25)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6e-9], 1.0, If[LessEqual[y, 2.15e-141], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.7e-104], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 3.7e-25], N[(x / z), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999998e-9 or 3.70000000000000009e-25 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 64.0%

      \[\leadsto \color{blue}{1} \]

   if -4.5999999999999998e-9 < y < 2.14999999999999987e-141 or 1.70000000000000008e-104 < y < 3.70000000000000009e-25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 66.4%

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

   if 2.14999999999999987e-141 < y < 1.70000000000000008e-104

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.9%

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

   4. Taylor expanded in z around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. distribute-frac-neg2 71.4%

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

   6. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{x}{-y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-104}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+20} \lor \neg \left(y \leq 3.7 \cdot 10^{-25}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+20) (not (<= y 3.7e-25))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+20) || !(y <= 3.7e-25)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.5d+20)) .or. (.not. (y <= 3.7d-25))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+20) || !(y <= 3.7e-25)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e+20) or not (y <= 3.7e-25):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e+20) || !(y <= 3.7e-25))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.5e+20) || ~((y <= 3.7e-25)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e+20], N[Not[LessEqual[y, 3.7e-25]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e20 or 3.70000000000000009e-25 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 80.8%

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

      1. div-sub 80.8%

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

      2. sub-neg 80.8%

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

      3. *-inverses 80.8%

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

      4. metadata-eval 80.8%

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

      5. distribute-lft-in 80.8%

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

      6. metadata-eval 80.8%

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

      7. +-commutative 80.8%

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

      8. mul-1-neg 80.8%

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

      9. unsub-neg 80.8%

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

   5. Simplified 80.8%

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

   if -3.5e20 < y < 3.70000000000000009e-25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.7%

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+20} \lor \neg \left(y \leq 3.7 \cdot 10^{-25}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e-9) 1.0 (if (<= y 4.2e-25) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-9) {
		tmp = 1.0;
	} else if (y <= 4.2e-25) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.8d-9)) then
        tmp = 1.0d0
    else if (y <= 4.2d-25) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-9) {
		tmp = 1.0;
	} else if (y <= 4.2e-25) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e-9:
		tmp = 1.0
	elif y <= 4.2e-25:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e-9)
		tmp = 1.0;
	elseif (y <= 4.2e-25)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e-9)
		tmp = 1.0;
	elseif (y <= 4.2e-25)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e-9], 1.0, If[LessEqual[y, 4.2e-25], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8e-9 or 4.20000000000000005e-25 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 64.0%

      \[\leadsto \color{blue}{1} \]

   if -4.8e-9 < y < 4.20000000000000005e-25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 63.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

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

Python

def code(x, y, z):
	return 1.0

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 37.1%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 37.1%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z - y} - \frac{y}{z - y} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / (z - y)) - (y / (z - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (- (/ x (- z y)) (/ y (- z y)))

  (/ (- x y) (- z y)))