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

Percentage Accurate: 99.8% → 100.0%

Time: 5.4s

Alternatives: 8

Speedup: 1.2×

• 21.0% 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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x - y}{z} - 2 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) / z)) - 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 = (4.0d0 * ((x - y) / z)) - 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - 2.0), $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/ 99.8%

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

   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 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-169}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-212}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-95}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x z))) (t_1 (* -4.0 (/ y z))))
   (if (<= x -1e+14)
     t_0
     (if (<= x -2.3e-100)
       t_1
       (if (<= x -5.3e-169)
         -2.0
         (if (<= x 1.75e-262)
           t_1
           (if (<= x 3.9e-212)
             -2.0
             (if (<= x 1.75e-158)
               t_1
               (if (<= x 4e-95) -2.0 (if (<= x 3.4e-31) 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 <= -1e+14) {
		tmp = t_0;
	} else if (x <= -2.3e-100) {
		tmp = t_1;
	} else if (x <= -5.3e-169) {
		tmp = -2.0;
	} else if (x <= 1.75e-262) {
		tmp = t_1;
	} else if (x <= 3.9e-212) {
		tmp = -2.0;
	} else if (x <= 1.75e-158) {
		tmp = t_1;
	} else if (x <= 4e-95) {
		tmp = -2.0;
	} else if (x <= 3.4e-31) {
		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 <= (-1d+14)) then
        tmp = t_0
    else if (x <= (-2.3d-100)) then
        tmp = t_1
    else if (x <= (-5.3d-169)) then
        tmp = -2.0d0
    else if (x <= 1.75d-262) then
        tmp = t_1
    else if (x <= 3.9d-212) then
        tmp = -2.0d0
    else if (x <= 1.75d-158) then
        tmp = t_1
    else if (x <= 4d-95) then
        tmp = -2.0d0
    else if (x <= 3.4d-31) 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 <= -1e+14) {
		tmp = t_0;
	} else if (x <= -2.3e-100) {
		tmp = t_1;
	} else if (x <= -5.3e-169) {
		tmp = -2.0;
	} else if (x <= 1.75e-262) {
		tmp = t_1;
	} else if (x <= 3.9e-212) {
		tmp = -2.0;
	} else if (x <= 1.75e-158) {
		tmp = t_1;
	} else if (x <= 4e-95) {
		tmp = -2.0;
	} else if (x <= 3.4e-31) {
		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 <= -1e+14:
		tmp = t_0
	elif x <= -2.3e-100:
		tmp = t_1
	elif x <= -5.3e-169:
		tmp = -2.0
	elif x <= 1.75e-262:
		tmp = t_1
	elif x <= 3.9e-212:
		tmp = -2.0
	elif x <= 1.75e-158:
		tmp = t_1
	elif x <= 4e-95:
		tmp = -2.0
	elif x <= 3.4e-31:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / z))
	t_1 = Float64(-4.0 * Float64(y / z))
	tmp = 0.0
	if (x <= -1e+14)
		tmp = t_0;
	elseif (x <= -2.3e-100)
		tmp = t_1;
	elseif (x <= -5.3e-169)
		tmp = -2.0;
	elseif (x <= 1.75e-262)
		tmp = t_1;
	elseif (x <= 3.9e-212)
		tmp = -2.0;
	elseif (x <= 1.75e-158)
		tmp = t_1;
	elseif (x <= 4e-95)
		tmp = -2.0;
	elseif (x <= 3.4e-31)
		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 <= -1e+14)
		tmp = t_0;
	elseif (x <= -2.3e-100)
		tmp = t_1;
	elseif (x <= -5.3e-169)
		tmp = -2.0;
	elseif (x <= 1.75e-262)
		tmp = t_1;
	elseif (x <= 3.9e-212)
		tmp = -2.0;
	elseif (x <= 1.75e-158)
		tmp = t_1;
	elseif (x <= 4e-95)
		tmp = -2.0;
	elseif (x <= 3.4e-31)
		tmp = t_1;
	else
		tmp = t_0;
	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[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+14], t$95$0, If[LessEqual[x, -2.3e-100], t$95$1, If[LessEqual[x, -5.3e-169], -2.0, If[LessEqual[x, 1.75e-262], t$95$1, If[LessEqual[x, 3.9e-212], -2.0, If[LessEqual[x, 1.75e-158], t$95$1, If[LessEqual[x, 4e-95], -2.0, If[LessEqual[x, 3.4e-31], t$95$1, t$95$0]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e14 or 3.4000000000000001e-31 < 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 65.8%

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

   if -1e14 < x < -2.29999999999999994e-100 or -5.3e-169 < x < 1.75000000000000006e-262 or 3.9e-212 < x < 1.75000000000000006e-158 or 3.99999999999999996e-95 < x < 3.4000000000000001e-31

   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. 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 y around inf 66.2%

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

   if -2.29999999999999994e-100 < x < -5.3e-169 or 1.75000000000000006e-262 < x < 3.9e-212 or 1.75000000000000006e-158 < x < 3.99999999999999996e-95

   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 72.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+14}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-100}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-169}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-262}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-212}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-95}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+65} \lor \neg \left(z \leq 2.2 \cdot 10^{+16}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6e+65) (not (<= z 2.2e+16)))
   (- (* -4.0 (/ y z)) 2.0)
   (* 4.0 (/ (- x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e+65) || !(z <= 2.2e+16)) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - 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 ((z <= (-6d+65)) .or. (.not. (z <= 2.2d+16))) then
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    else
        tmp = 4.0d0 * ((x - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e+65) || !(z <= 2.2e+16)) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6e+65) or not (z <= 2.2e+16):
		tmp = (-4.0 * (y / z)) - 2.0
	else:
		tmp = 4.0 * ((x - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6e+65) || !(z <= 2.2e+16))
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	else
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6e+65) || ~((z <= 2.2e+16)))
		tmp = (-4.0 * (y / z)) - 2.0;
	else
		tmp = 4.0 * ((x - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6e+65], N[Not[LessEqual[z, 2.2e+16]], $MachinePrecision]], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.0000000000000004e65 or 2.2e16 < 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/ 99.8%

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

      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-lft-in 99.7%

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

      4. associate-*l/ 99.9%

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

      5. associate-*r/ 99.9%

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

      6. fma-def 99.9%

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

      7. associate-*l/ 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. associate-*r* 100.0%

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

      11. remove-double-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{2}}{-1} \cdot \frac{-z}{-z}\right) \]

      15. metadata-eval 100.0%

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

      16. *-inverses 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, -2 \cdot \color{blue}{1}\right) \]

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 81.3%

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

   if -6.0000000000000004e65 < z < 2.2e16

   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. 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 0 93.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+65} \lor \neg \left(z \leq 2.2 \cdot 10^{+16}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+172}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+170}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+172) -2.0 (if (<= z 2.3e+170) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+172) {
		tmp = -2.0;
	} else if (z <= 2.3e+170) {
		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 <= (-2.3d+172)) then
        tmp = -2.0d0
    else if (z <= 2.3d+170) 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 <= -2.3e+172) {
		tmp = -2.0;
	} else if (z <= 2.3e+170) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.3e+172:
		tmp = -2.0
	elif z <= 2.3e+170:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+172)
		tmp = -2.0;
	elseif (z <= 2.3e+170)
		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 <= -2.3e+172)
		tmp = -2.0;
	elseif (z <= 2.3e+170)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+172], -2.0, If[LessEqual[z, 2.3e+170], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e172 or 2.3000000000000001e170 < 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/ 99.8%

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

      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 87.2%

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

   if -2.3000000000000001e172 < z < 2.3000000000000001e170

   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. 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%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+172}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+170}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0085:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0085)
   (* 4.0 (/ (- x y) z))
   (if (<= x 4.2e-36) (- (* -4.0 (/ y z)) 2.0) (- (* 4.0 (/ x z)) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0085) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 4.2e-36) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / 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 <= (-0.0085d0)) then
        tmp = 4.0d0 * ((x - y) / z)
    else if (x <= 4.2d-36) then
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    else
        tmp = (4.0d0 * (x / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0085) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 4.2e-36) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.0085:
		tmp = 4.0 * ((x - y) / z)
	elif x <= 4.2e-36:
		tmp = (-4.0 * (y / z)) - 2.0
	else:
		tmp = (4.0 * (x / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0085)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	elseif (x <= 4.2e-36)
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(x / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.0085)
		tmp = 4.0 * ((x - y) / z);
	elseif (x <= 4.2e-36)
		tmp = (-4.0 * (y / z)) - 2.0;
	else
		tmp = (4.0 * (x / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0085], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-36], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0085000000000000006

   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. 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 84.6%

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

   if -0.0085000000000000006 < x < 4.19999999999999982e-36

   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. 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-lft-in 99.8%

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

      4. associate-*l/ 99.9%

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

      5. associate-*r/ 99.9%

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

      6. fma-def 99.9%

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

      7. associate-*l/ 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. associate-*r* 100.0%

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

      11. remove-double-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{2}}{-1} \cdot \frac{-z}{-z}\right) \]

      15. metadata-eval 100.0%

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

      16. *-inverses 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, -2 \cdot \color{blue}{1}\right) \]

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 94.8%

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

   if 4.19999999999999982e-36 < 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/ 99.8%

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

      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-lft-in 99.8%

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

      4. associate-*l/ 99.9%

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

      5. associate-*r/ 99.9%

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

      6. fma-def 99.9%

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

      7. associate-*l/ 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. associate-*r* 100.0%

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

      11. remove-double-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{2}}{-1} \cdot \frac{-z}{-z}\right) \]

      15. metadata-eval 100.0%

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

      16. *-inverses 100.0%

          \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, -2 \cdot \color{blue}{1}\right) \]

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 90.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0085:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \]

Alternative 6: 52.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-21} \lor \neg \left(y \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8e-21) (not (<= y 2.6e+77))) (* -4.0 (/ y z)) -2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e-21) || !(y <= 2.6e+77)) {
		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 <= (-8d-21)) .or. (.not. (y <= 2.6d+77))) 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 <= -8e-21) || !(y <= 2.6e+77)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8e-21) or not (y <= 2.6e+77):
		tmp = -4.0 * (y / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8e-21) || !(y <= 2.6e+77))
		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 <= -8e-21) || ~((y <= 2.6e+77)))
		tmp = -4.0 * (y / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8e-21], N[Not[LessEqual[y, 2.6e+77]], $MachinePrecision]], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999926e-21 or 2.6000000000000002e77 < 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/ 99.8%

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

      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 y around inf 61.2%

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

   if -7.99999999999999926e-21 < y < 2.6000000000000002e77

   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. 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 43.8%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-21} \lor \neg \left(y \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((x - y) * (4.0 / z)) + -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 = ((x - y) * (4.0d0 / z)) + (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision]), $MachinePrecision] + -2.0), $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/ 99.8%

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

   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-lft-in 99.8%

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

   4. associate-*l/ 99.9%

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

   5. associate-*r/ 99.9%

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

   6. fma-def 99.9%

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

   7. associate-*l/ 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. associate-*r* 100.0%

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

   11. remove-double-neg 100.0%

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

   12. neg-mul-1 100.0%

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

   13. times-frac 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{2}}{-1} \cdot \frac{-z}{-z}\right) \]

   15. metadata-eval 100.0%

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

   16. *-inverses 100.0%

       \[\leadsto \mathsf{fma}\left(4, \frac{x - y}{z}, -2 \cdot \color{blue}{1}\right) \]

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. associate-*r/ 100.0%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z}} + -2 \]

   3. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - y\right)} + -2 \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - y\right) + -2} \]

6. Final simplification 99.8%

   \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + -2 \]

Alternative 8: 34.1% 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/ 99.8%

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

   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 30.4%

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

5. Final simplification 30.4%

   \[\leadsto -2 \]

Developer target: 97.9% 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 2023238 
(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))