Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
4. Add Preprocessing

Alternative 2: 53.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-148}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-265}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-121}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 z))) (t_1 (* -4.0 (/ y z))))
   (if (<= x -3.8e+51)
     t_0
     (if (<= x -4.4e-127)
       t_1
       (if (<= x -2.55e-148)
         -2.0
         (if (<= x -4.8e-180)
           t_1
           (if (<= x 9.5e-265)
             -2.0
             (if (<= x 1.35e-167)
               t_1
               (if (<= x 2.05e-121) -2.0 (if (<= x 1.12e-23) t_1 t_0))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / z);
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -3.8e+51) {
		tmp = t_0;
	} else if (x <= -4.4e-127) {
		tmp = t_1;
	} else if (x <= -2.55e-148) {
		tmp = -2.0;
	} else if (x <= -4.8e-180) {
		tmp = t_1;
	} else if (x <= 9.5e-265) {
		tmp = -2.0;
	} else if (x <= 1.35e-167) {
		tmp = t_1;
	} else if (x <= 2.05e-121) {
		tmp = -2.0;
	} else if (x <= 1.12e-23) {
		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 = x * (4.0d0 / z)
    t_1 = (-4.0d0) * (y / z)
    if (x <= (-3.8d+51)) then
        tmp = t_0
    else if (x <= (-4.4d-127)) then
        tmp = t_1
    else if (x <= (-2.55d-148)) then
        tmp = -2.0d0
    else if (x <= (-4.8d-180)) then
        tmp = t_1
    else if (x <= 9.5d-265) then
        tmp = -2.0d0
    else if (x <= 1.35d-167) then
        tmp = t_1
    else if (x <= 2.05d-121) then
        tmp = -2.0d0
    else if (x <= 1.12d-23) 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 = x * (4.0 / z);
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -3.8e+51) {
		tmp = t_0;
	} else if (x <= -4.4e-127) {
		tmp = t_1;
	} else if (x <= -2.55e-148) {
		tmp = -2.0;
	} else if (x <= -4.8e-180) {
		tmp = t_1;
	} else if (x <= 9.5e-265) {
		tmp = -2.0;
	} else if (x <= 1.35e-167) {
		tmp = t_1;
	} else if (x <= 2.05e-121) {
		tmp = -2.0;
	} else if (x <= 1.12e-23) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / z)
	t_1 = -4.0 * (y / z)
	tmp = 0
	if x <= -3.8e+51:
		tmp = t_0
	elif x <= -4.4e-127:
		tmp = t_1
	elif x <= -2.55e-148:
		tmp = -2.0
	elif x <= -4.8e-180:
		tmp = t_1
	elif x <= 9.5e-265:
		tmp = -2.0
	elif x <= 1.35e-167:
		tmp = t_1
	elif x <= 2.05e-121:
		tmp = -2.0
	elif x <= 1.12e-23:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / z))
	t_1 = Float64(-4.0 * Float64(y / z))
	tmp = 0.0
	if (x <= -3.8e+51)
		tmp = t_0;
	elseif (x <= -4.4e-127)
		tmp = t_1;
	elseif (x <= -2.55e-148)
		tmp = -2.0;
	elseif (x <= -4.8e-180)
		tmp = t_1;
	elseif (x <= 9.5e-265)
		tmp = -2.0;
	elseif (x <= 1.35e-167)
		tmp = t_1;
	elseif (x <= 2.05e-121)
		tmp = -2.0;
	elseif (x <= 1.12e-23)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (4.0 / z);
	t_1 = -4.0 * (y / z);
	tmp = 0.0;
	if (x <= -3.8e+51)
		tmp = t_0;
	elseif (x <= -4.4e-127)
		tmp = t_1;
	elseif (x <= -2.55e-148)
		tmp = -2.0;
	elseif (x <= -4.8e-180)
		tmp = t_1;
	elseif (x <= 9.5e-265)
		tmp = -2.0;
	elseif (x <= 1.35e-167)
		tmp = t_1;
	elseif (x <= 2.05e-121)
		tmp = -2.0;
	elseif (x <= 1.12e-23)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+51], t$95$0, If[LessEqual[x, -4.4e-127], t$95$1, If[LessEqual[x, -2.55e-148], -2.0, If[LessEqual[x, -4.8e-180], t$95$1, If[LessEqual[x, 9.5e-265], -2.0, If[LessEqual[x, 1.35e-167], t$95$1, If[LessEqual[x, 2.05e-121], -2.0, If[LessEqual[x, 1.12e-23], t$95$1, t$95$0]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999997e51 or 1.1200000000000001e-23 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot 4} \]

      2. associate-*l/ 67.1%

         \[\leadsto \color{blue}{\frac{x \cdot 4}{z}} \]

      3. associate-*r/ 67.0%

         \[\leadsto \color{blue}{x \cdot \frac{4}{z}} \]

   5. Simplified 67.0%

      \[\leadsto \color{blue}{x \cdot \frac{4}{z}} \]

   if -3.7999999999999997e51 < x < -4.4000000000000003e-127 or -2.55e-148 < x < -4.79999999999999959e-180 or 9.50000000000000034e-265 < x < 1.35e-167 or 2.04999999999999991e-121 < x < 1.1200000000000001e-23

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]

   if -4.4000000000000003e-127 < x < -2.55e-148 or -4.79999999999999959e-180 < x < 9.50000000000000034e-265 or 1.35e-167 < x < 2.04999999999999991e-121

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 76.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-127}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-148}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-180}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-265}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-167}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-121}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-23}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-149}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-265}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) z)) (t_1 (* -4.0 (/ y z))))
   (if (<= x -1.5e+55)
     t_0
     (if (<= x -3.9e-125)
       t_1
       (if (<= x -4.8e-149)
         -2.0
         (if (<= x -1.2e-179)
           t_1
           (if (<= x 3.6e-265)
             -2.0
             (if (<= x 6.4e-167)
               t_1
               (if (<= x 3.9e-119) -2.0 (if (<= x 1.12e-23) t_1 t_0))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / z;
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -1.5e+55) {
		tmp = t_0;
	} else if (x <= -3.9e-125) {
		tmp = t_1;
	} else if (x <= -4.8e-149) {
		tmp = -2.0;
	} else if (x <= -1.2e-179) {
		tmp = t_1;
	} else if (x <= 3.6e-265) {
		tmp = -2.0;
	} else if (x <= 6.4e-167) {
		tmp = t_1;
	} else if (x <= 3.9e-119) {
		tmp = -2.0;
	} else if (x <= 1.12e-23) {
		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 = (4.0d0 * x) / z
    t_1 = (-4.0d0) * (y / z)
    if (x <= (-1.5d+55)) then
        tmp = t_0
    else if (x <= (-3.9d-125)) then
        tmp = t_1
    else if (x <= (-4.8d-149)) then
        tmp = -2.0d0
    else if (x <= (-1.2d-179)) then
        tmp = t_1
    else if (x <= 3.6d-265) then
        tmp = -2.0d0
    else if (x <= 6.4d-167) then
        tmp = t_1
    else if (x <= 3.9d-119) then
        tmp = -2.0d0
    else if (x <= 1.12d-23) 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 = (4.0 * x) / z;
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -1.5e+55) {
		tmp = t_0;
	} else if (x <= -3.9e-125) {
		tmp = t_1;
	} else if (x <= -4.8e-149) {
		tmp = -2.0;
	} else if (x <= -1.2e-179) {
		tmp = t_1;
	} else if (x <= 3.6e-265) {
		tmp = -2.0;
	} else if (x <= 6.4e-167) {
		tmp = t_1;
	} else if (x <= 3.9e-119) {
		tmp = -2.0;
	} else if (x <= 1.12e-23) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * x) / z
	t_1 = -4.0 * (y / z)
	tmp = 0
	if x <= -1.5e+55:
		tmp = t_0
	elif x <= -3.9e-125:
		tmp = t_1
	elif x <= -4.8e-149:
		tmp = -2.0
	elif x <= -1.2e-179:
		tmp = t_1
	elif x <= 3.6e-265:
		tmp = -2.0
	elif x <= 6.4e-167:
		tmp = t_1
	elif x <= 3.9e-119:
		tmp = -2.0
	elif x <= 1.12e-23:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / z)
	t_1 = Float64(-4.0 * Float64(y / z))
	tmp = 0.0
	if (x <= -1.5e+55)
		tmp = t_0;
	elseif (x <= -3.9e-125)
		tmp = t_1;
	elseif (x <= -4.8e-149)
		tmp = -2.0;
	elseif (x <= -1.2e-179)
		tmp = t_1;
	elseif (x <= 3.6e-265)
		tmp = -2.0;
	elseif (x <= 6.4e-167)
		tmp = t_1;
	elseif (x <= 3.9e-119)
		tmp = -2.0;
	elseif (x <= 1.12e-23)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * x) / z;
	t_1 = -4.0 * (y / z);
	tmp = 0.0;
	if (x <= -1.5e+55)
		tmp = t_0;
	elseif (x <= -3.9e-125)
		tmp = t_1;
	elseif (x <= -4.8e-149)
		tmp = -2.0;
	elseif (x <= -1.2e-179)
		tmp = t_1;
	elseif (x <= 3.6e-265)
		tmp = -2.0;
	elseif (x <= 6.4e-167)
		tmp = t_1;
	elseif (x <= 3.9e-119)
		tmp = -2.0;
	elseif (x <= 1.12e-23)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e+55], t$95$0, If[LessEqual[x, -3.9e-125], t$95$1, If[LessEqual[x, -4.8e-149], -2.0, If[LessEqual[x, -1.2e-179], t$95$1, If[LessEqual[x, 3.6e-265], -2.0, If[LessEqual[x, 6.4e-167], t$95$1, If[LessEqual[x, 3.9e-119], -2.0, If[LessEqual[x, 1.12e-23], t$95$1, t$95$0]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000008e55 or 1.1200000000000001e-23 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.1%

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

      1. associate-*r/ 67.1%

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

   if -1.50000000000000008e55 < x < -3.89999999999999982e-125 or -4.8000000000000002e-149 < x < -1.2e-179 or 3.6000000000000002e-265 < x < 6.4000000000000003e-167 or 3.8999999999999999e-119 < x < 1.1200000000000001e-23

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]

   if -3.89999999999999982e-125 < x < -4.8000000000000002e-149 or -1.2e-179 < x < 3.6000000000000002e-265 or 6.4000000000000003e-167 < x < 3.8999999999999999e-119

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 76.8%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-125}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-149}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-179}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-265}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-167}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-23}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+52} \lor \neg \left(x \leq 270000000\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e+52) (not (<= x 270000000.0)))
   (* 4.0 (/ (- x y) z))
   (- (* -4.0 (/ y z)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+52) || !(x <= 270000000.0)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	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 ((x <= (-2d+52)) .or. (.not. (x <= 270000000.0d0))) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+52) || !(x <= 270000000.0)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e+52) or not (x <= 270000000.0):
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = (-4.0 * (y / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e+52) || !(x <= 270000000.0))
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e+52) || ~((x <= 270000000.0)))
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = (-4.0 * (y / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e+52], N[Not[LessEqual[x, 270000000.0]], $MachinePrecision]], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e52 or 2.7e8 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 83.5%

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]

   if -2e52 < x < 2.7e8

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 89.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+52} \lor \neg \left(x \leq 270000000\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+158}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+158) -2.0 (if (<= z 6.4e+125) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+158) {
		tmp = -2.0;
	} else if (z <= 6.4e+125) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	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 (z <= (-3.1d+158)) then
        tmp = -2.0d0
    else if (z <= 6.4d+125) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+158) {
		tmp = -2.0;
	} else if (z <= 6.4e+125) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e+158:
		tmp = -2.0
	elif z <= 6.4e+125:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+158)
		tmp = -2.0;
	elseif (z <= 6.4e+125)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.1e+158)
		tmp = -2.0;
	elseif (z <= 6.4e+125)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+158], -2.0, If[LessEqual[z, 6.4e+125], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000002e158 or 6.39999999999999967e125 < z

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 79.9%

      \[\leadsto \color{blue}{-2} \]

   if -3.1000000000000002e158 < z < 6.39999999999999967e125

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+158}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-56}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e-56)
   (* 4.0 (/ (- x y) z))
   (if (<= y 1.45e+15) (- (* 4.0 (/ x z)) 2.0) (- (* -4.0 (/ y z)) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e-56) {
		tmp = 4.0 * ((x - y) / z);
	} else if (y <= 1.45e+15) {
		tmp = (4.0 * (x / z)) - 2.0;
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	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 <= (-2.2d-56)) then
        tmp = 4.0d0 * ((x - y) / z)
    else if (y <= 1.45d+15) then
        tmp = (4.0d0 * (x / z)) - 2.0d0
    else
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e-56) {
		tmp = 4.0 * ((x - y) / z);
	} else if (y <= 1.45e+15) {
		tmp = (4.0 * (x / z)) - 2.0;
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e-56:
		tmp = 4.0 * ((x - y) / z)
	elif y <= 1.45e+15:
		tmp = (4.0 * (x / z)) - 2.0
	else:
		tmp = (-4.0 * (y / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e-56)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	elseif (y <= 1.45e+15)
		tmp = Float64(Float64(4.0 * Float64(x / z)) - 2.0);
	else
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e-56)
		tmp = 4.0 * ((x - y) / z);
	elseif (y <= 1.45e+15)
		tmp = (4.0 * (x / z)) - 2.0;
	else
		tmp = (-4.0 * (y / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e-56], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+15], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000004e-56

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]

   if -2.20000000000000004e-56 < y < 1.45e15

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 95.5%

      \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - 2} \]

   if 1.45e15 < y

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 99.9%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 99.9%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 89.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-56}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-56} \lor \neg \left(y \leq 5.5 \cdot 10^{+58}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e-56) (not (<= y 5.5e+58))) (* -4.0 (/ y z)) -2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-56) || !(y <= 5.5e+58)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	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 <= (-4d-56)) .or. (.not. (y <= 5.5d+58))) then
        tmp = (-4.0d0) * (y / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-56) || !(y <= 5.5e+58)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e-56) or not (y <= 5.5e+58):
		tmp = -4.0 * (y / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e-56) || !(y <= 5.5e+58))
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e-56) || ~((y <= 5.5e+58)))
		tmp = -4.0 * (y / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e-56], N[Not[LessEqual[y, 5.5e+58]], $MachinePrecision]], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000002e-56 or 5.4999999999999999e58 < y

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 63.4%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]

   if -4.0000000000000002e-56 < y < 5.4999999999999999e58

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 50.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-56} \lor \neg \left(y \leq 5.5 \cdot 10^{+58}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 34.3%

   \[\leadsto \color{blue}{-2} \]

4. Final simplification 34.3%

   \[\leadsto -2 \]
5. Add Preprocessing

Developer target: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024010 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))