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

Percentage Accurate: 99.8% → 99.8%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

• 20.8% 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: 99.8% 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: 99.8% 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. Final simplification 100.0%

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

Alternative 2: 52.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := \frac{y \cdot -4}{z}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x z))) (t_1 (/ (* y -4.0) z)))
   (if (<= y -1.05e+47)
     t_1
     (if (<= y -5.8e-143)
       -2.0
       (if (<= y 1.75e-173)
         t_0
         (if (<= y 2.75e-132) -2.0 (if (<= y 9.5e+165) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / z);
	double t_1 = (y * -4.0) / z;
	double tmp;
	if (y <= -1.05e+47) {
		tmp = t_1;
	} else if (y <= -5.8e-143) {
		tmp = -2.0;
	} else if (y <= 1.75e-173) {
		tmp = t_0;
	} else if (y <= 2.75e-132) {
		tmp = -2.0;
	} else if (y <= 9.5e+165) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = (y * (-4.0d0)) / z
    if (y <= (-1.05d+47)) then
        tmp = t_1
    else if (y <= (-5.8d-143)) then
        tmp = -2.0d0
    else if (y <= 1.75d-173) then
        tmp = t_0
    else if (y <= 2.75d-132) then
        tmp = -2.0d0
    else if (y <= 9.5d+165) then
        tmp = t_0
    else
        tmp = t_1
    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 = (y * -4.0) / z;
	double tmp;
	if (y <= -1.05e+47) {
		tmp = t_1;
	} else if (y <= -5.8e-143) {
		tmp = -2.0;
	} else if (y <= 1.75e-173) {
		tmp = t_0;
	} else if (y <= 2.75e-132) {
		tmp = -2.0;
	} else if (y <= 9.5e+165) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / z)
	t_1 = (y * -4.0) / z
	tmp = 0
	if y <= -1.05e+47:
		tmp = t_1
	elif y <= -5.8e-143:
		tmp = -2.0
	elif y <= 1.75e-173:
		tmp = t_0
	elif y <= 2.75e-132:
		tmp = -2.0
	elif y <= 9.5e+165:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / z))
	t_1 = Float64(Float64(y * -4.0) / z)
	tmp = 0.0
	if (y <= -1.05e+47)
		tmp = t_1;
	elseif (y <= -5.8e-143)
		tmp = -2.0;
	elseif (y <= 1.75e-173)
		tmp = t_0;
	elseif (y <= 2.75e-132)
		tmp = -2.0;
	elseif (y <= 9.5e+165)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / z);
	t_1 = (y * -4.0) / z;
	tmp = 0.0;
	if (y <= -1.05e+47)
		tmp = t_1;
	elseif (y <= -5.8e-143)
		tmp = -2.0;
	elseif (y <= 1.75e-173)
		tmp = t_0;
	elseif (y <= 2.75e-132)
		tmp = -2.0;
	elseif (y <= 9.5e+165)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.05e+47], t$95$1, If[LessEqual[y, -5.8e-143], -2.0, If[LessEqual[y, 1.75e-173], t$95$0, If[LessEqual[y, 2.75e-132], -2.0, If[LessEqual[y, 9.5e+165], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e47 or 9.50000000000000017e165 < y

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l/ 77.2%

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

   6. Simplified 77.2%

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

   if -1.05e47 < y < -5.8000000000000002e-143 or 1.75000000000000007e-173 < y < 2.75e-132

   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. associate-*l/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 76.5%

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

   if -5.8000000000000002e-143 < y < 1.75000000000000007e-173 or 2.75e-132 < y < 9.50000000000000017e165

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 56.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-173}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+165}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \end{array} \]

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+48}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 12500000 \lor \neg \left(y \leq 1.06 \cdot 10^{+57}\right) \land y \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.2e+48)
   (* 4.0 (- -0.5 (/ y z)))
   (if (or (<= y 12500000.0) (and (not (<= y 1.06e+57)) (<= y 4.2e+109)))
     (+ (* 4.0 (/ x z)) -2.0)
     (* (- x y) (/ 4.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+48) {
		tmp = 4.0 * (-0.5 - (y / z));
	} else if ((y <= 12500000.0) || (!(y <= 1.06e+57) && (y <= 4.2e+109))) {
		tmp = (4.0 * (x / z)) + -2.0;
	} else {
		tmp = (x - y) * (4.0 / 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) :: tmp
    if (y <= (-8.2d+48)) then
        tmp = 4.0d0 * ((-0.5d0) - (y / z))
    else if ((y <= 12500000.0d0) .or. (.not. (y <= 1.06d+57)) .and. (y <= 4.2d+109)) then
        tmp = (4.0d0 * (x / z)) + (-2.0d0)
    else
        tmp = (x - y) * (4.0d0 / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+48) {
		tmp = 4.0 * (-0.5 - (y / z));
	} else if ((y <= 12500000.0) || (!(y <= 1.06e+57) && (y <= 4.2e+109))) {
		tmp = (4.0 * (x / z)) + -2.0;
	} else {
		tmp = (x - y) * (4.0 / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.2e+48:
		tmp = 4.0 * (-0.5 - (y / z))
	elif (y <= 12500000.0) or (not (y <= 1.06e+57) and (y <= 4.2e+109)):
		tmp = (4.0 * (x / z)) + -2.0
	else:
		tmp = (x - y) * (4.0 / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.2e+48)
		tmp = Float64(4.0 * Float64(-0.5 - Float64(y / z)));
	elseif ((y <= 12500000.0) || (!(y <= 1.06e+57) && (y <= 4.2e+109)))
		tmp = Float64(Float64(4.0 * Float64(x / z)) + -2.0);
	else
		tmp = Float64(Float64(x - y) * Float64(4.0 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.2e+48)
		tmp = 4.0 * (-0.5 - (y / z));
	elseif ((y <= 12500000.0) || (~((y <= 1.06e+57)) && (y <= 4.2e+109)))
		tmp = (4.0 * (x / z)) + -2.0;
	else
		tmp = (x - y) * (4.0 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.2e+48], N[(4.0 * N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 12500000.0], And[N[Not[LessEqual[y, 1.06e+57]], $MachinePrecision], LessEqual[y, 4.2e+109]]], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000005e48

   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. associate-*l/ 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{4 \cdot \frac{-0.5 \cdot z - y}{z}} \]
   5. Step-by-step derivation

      1. div-sub 85.3%

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

      2. *-commutative 85.3%

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

      3. *-lft-identity 85.3%

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

      4. associate-*l/ 85.2%

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

      5. associate-*r* 85.2%

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

      6. lft-mult-inverse 85.3%

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

      7. metadata-eval 85.3%

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

   6. Simplified 85.3%

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

   if -8.2000000000000005e48 < y < 1.25e7 or 1.06e57 < y < 4.2000000000000003e109

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 95.5%

      \[\leadsto \color{blue}{4 \cdot \frac{x + -0.5 \cdot z}{z}} \]
   5. Step-by-step derivation

      1. metadata-eval 95.5%

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

      2. cancel-sign-sub-inv 95.5%

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

      3. div-sub 95.5%

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

      4. sub-neg 95.5%

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

      5. associate-/l* 95.5%

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

      6. *-lft-identity 95.5%

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

      7. associate-*l/ 95.4%

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

      8. lft-mult-inverse 95.5%

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

      9. metadata-eval 95.5%

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

      10. metadata-eval 95.5%

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

      11. distribute-lft-in 95.5%

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

      12. metadata-eval 95.5%

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

   6. Simplified 95.5%

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

   if 1.25e7 < y < 1.06e57 or 4.2000000000000003e109 < y

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 93.8%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l/ 93.6%

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

      3. *-commutative 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+48}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 12500000 \lor \neg \left(y \leq 1.06 \cdot 10^{+57}\right) \land y \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+83} \lor \neg \left(x \leq 3.6 \cdot 10^{+82}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.15e+83) (not (<= x 3.6e+82)))
   (* 4.0 (/ x z))
   (* 4.0 (- -0.5 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.15e+83) || !(x <= 3.6e+82)) {
		tmp = 4.0 * (x / z);
	} else {
		tmp = 4.0 * (-0.5 - (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) :: tmp
    if ((x <= (-2.15d+83)) .or. (.not. (x <= 3.6d+82))) then
        tmp = 4.0d0 * (x / z)
    else
        tmp = 4.0d0 * ((-0.5d0) - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.15e+83) || !(x <= 3.6e+82)) {
		tmp = 4.0 * (x / z);
	} else {
		tmp = 4.0 * (-0.5 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.15e+83) or not (x <= 3.6e+82):
		tmp = 4.0 * (x / z)
	else:
		tmp = 4.0 * (-0.5 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.15e+83) || !(x <= 3.6e+82))
		tmp = Float64(4.0 * Float64(x / z));
	else
		tmp = Float64(4.0 * Float64(-0.5 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.15e+83) || ~((x <= 3.6e+82)))
		tmp = 4.0 * (x / z);
	else
		tmp = 4.0 * (-0.5 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.15e+83], N[Not[LessEqual[x, 3.6e+82]], $MachinePrecision]], N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15e83 or 3.60000000000000014e82 < x

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 77.7%

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

   if -2.15e83 < x < 3.60000000000000014e82

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 87.6%

      \[\leadsto \color{blue}{4 \cdot \frac{-0.5 \cdot z - y}{z}} \]
   5. Step-by-step derivation

      1. div-sub 87.6%

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

      2. *-commutative 87.6%

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

      3. *-lft-identity 87.6%

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

      4. associate-*l/ 87.4%

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

      5. associate-*r* 87.4%

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

      6. lft-mult-inverse 87.6%

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

      7. metadata-eval 87.6%

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

   6. Simplified 87.6%

      \[\leadsto \color{blue}{4 \cdot \left(-0.5 - \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+83} \lor \neg \left(x \leq 3.6 \cdot 10^{+82}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+35} \lor \neg \left(x \leq 7.4 \cdot 10^{+33}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.6e+35) (not (<= x 7.4e+33)))
   (* (- x y) (/ 4.0 z))
   (* 4.0 (- -0.5 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.6e+35) || !(x <= 7.4e+33)) {
		tmp = (x - y) * (4.0 / z);
	} else {
		tmp = 4.0 * (-0.5 - (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) :: tmp
    if ((x <= (-5.6d+35)) .or. (.not. (x <= 7.4d+33))) then
        tmp = (x - y) * (4.0d0 / z)
    else
        tmp = 4.0d0 * ((-0.5d0) - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.6e+35) || !(x <= 7.4e+33)) {
		tmp = (x - y) * (4.0 / z);
	} else {
		tmp = 4.0 * (-0.5 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.6e+35) or not (x <= 7.4e+33):
		tmp = (x - y) * (4.0 / z)
	else:
		tmp = 4.0 * (-0.5 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.6e+35) || !(x <= 7.4e+33))
		tmp = Float64(Float64(x - y) * Float64(4.0 / z));
	else
		tmp = Float64(4.0 * Float64(-0.5 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.6e+35) || ~((x <= 7.4e+33)))
		tmp = (x - y) * (4.0 / z);
	else
		tmp = 4.0 * (-0.5 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.6e+35], N[Not[LessEqual[x, 7.4e+33]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999997e35 or 7.3999999999999997e33 < x

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-*l/ 83.1%

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

      3. *-commutative 83.1%

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

   6. Simplified 83.1%

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

   if -5.59999999999999997e35 < x < 7.3999999999999997e33

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 90.6%

      \[\leadsto \color{blue}{4 \cdot \frac{-0.5 \cdot z - y}{z}} \]
   5. Step-by-step derivation

      1. div-sub 90.6%

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

      2. *-commutative 90.6%

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

      3. *-lft-identity 90.6%

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

      4. associate-*l/ 90.5%

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

      5. associate-*r* 90.5%

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

      6. lft-mult-inverse 90.6%

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

      7. metadata-eval 90.6%

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

   6. Simplified 90.6%

      \[\leadsto \color{blue}{4 \cdot \left(-0.5 - \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+35} \lor \neg \left(x \leq 7.4 \cdot 10^{+33}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 4.0 * ((-0.5 + (x / z)) - (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 * (((-0.5d0) + (x / z)) - (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   2. sub-neg 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \frac{-0.5 \cdot z - y}{z}} \]
5. Step-by-step derivation

   1. distribute-lft-out 97.6%

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

   2. div-sub 97.6%

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

   3. associate-+r- 97.6%

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

   4. *-commutative 97.6%

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

   5. *-lft-identity 97.6%

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

   6. associate-*l/ 97.6%

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

   7. associate-*r* 97.6%

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

   8. lft-mult-inverse 97.6%

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

   9. metadata-eval 97.6%

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

   10. +-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 7: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / z) * ((x - y) + (z * (-0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   2. sub-neg 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 8: 54.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+76}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.4e+76) -2.0 (if (<= z 2.8e+55) (* 4.0 (/ x z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.4e+76) {
		tmp = -2.0;
	} else if (z <= 2.8e+55) {
		tmp = 4.0 * (x / 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 <= (-4.4d+76)) then
        tmp = -2.0d0
    else if (z <= 2.8d+55) then
        tmp = 4.0d0 * (x / 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 <= -4.4e+76) {
		tmp = -2.0;
	} else if (z <= 2.8e+55) {
		tmp = 4.0 * (x / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.4e+76:
		tmp = -2.0
	elif z <= 2.8e+55:
		tmp = 4.0 * (x / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.4e+76)
		tmp = -2.0;
	elseif (z <= 2.8e+55)
		tmp = Float64(4.0 * Float64(x / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.4e+76)
		tmp = -2.0;
	elseif (z <= 2.8e+55)
		tmp = 4.0 * (x / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.4e+76], -2.0, If[LessEqual[z, 2.8e+55], N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000001e76 or 2.8000000000000001e55 < z

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 67.2%

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

   if -4.4000000000000001e76 < z < 2.8000000000000001e55

   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. associate-*l/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 50.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+76}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 9: 34.4% 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. Step-by-step derivation

   1. associate-*l/ 99.7%

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

   2. sub-neg 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in z around inf 33.4%

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

5. Final simplification 33.4%

   \[\leadsto -2 \]

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 2023271 
(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))