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

Percentage Accurate: 99.7% → 99.9%

Time: 6.8s

Alternatives: 11

Speedup: 1.4×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.8%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. +-commutative 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   3. fma-def 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

4. Taylor expanded in y around 0 100.0%

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

   1. associate-*r/ 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 53.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := 1 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -13600000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-120}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ z y))) (t_1 (+ 1.0 (* 4.0 (/ x y)))))
   (if (<= x -13600000.0)
     t_1
     (if (<= x -8e-57)
       t_0
       (if (<= x 4.4e-120)
         4.0
         (if (<= x 2.6e-57)
           t_0
           (if (<= x 1.16e+17) 4.0 (if (<= x 9e+107) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (z / y);
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -13600000.0) {
		tmp = t_1;
	} else if (x <= -8e-57) {
		tmp = t_0;
	} else if (x <= 4.4e-120) {
		tmp = 4.0;
	} else if (x <= 2.6e-57) {
		tmp = t_0;
	} else if (x <= 1.16e+17) {
		tmp = 4.0;
	} else if (x <= 9e+107) {
		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) * (z / y)
    t_1 = 1.0d0 + (4.0d0 * (x / y))
    if (x <= (-13600000.0d0)) then
        tmp = t_1
    else if (x <= (-8d-57)) then
        tmp = t_0
    else if (x <= 4.4d-120) then
        tmp = 4.0d0
    else if (x <= 2.6d-57) then
        tmp = t_0
    else if (x <= 1.16d+17) then
        tmp = 4.0d0
    else if (x <= 9d+107) 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 * (z / y);
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -13600000.0) {
		tmp = t_1;
	} else if (x <= -8e-57) {
		tmp = t_0;
	} else if (x <= 4.4e-120) {
		tmp = 4.0;
	} else if (x <= 2.6e-57) {
		tmp = t_0;
	} else if (x <= 1.16e+17) {
		tmp = 4.0;
	} else if (x <= 9e+107) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (z / y)
	t_1 = 1.0 + (4.0 * (x / y))
	tmp = 0
	if x <= -13600000.0:
		tmp = t_1
	elif x <= -8e-57:
		tmp = t_0
	elif x <= 4.4e-120:
		tmp = 4.0
	elif x <= 2.6e-57:
		tmp = t_0
	elif x <= 1.16e+17:
		tmp = 4.0
	elif x <= 9e+107:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(z / y))
	t_1 = Float64(1.0 + Float64(4.0 * Float64(x / y)))
	tmp = 0.0
	if (x <= -13600000.0)
		tmp = t_1;
	elseif (x <= -8e-57)
		tmp = t_0;
	elseif (x <= 4.4e-120)
		tmp = 4.0;
	elseif (x <= 2.6e-57)
		tmp = t_0;
	elseif (x <= 1.16e+17)
		tmp = 4.0;
	elseif (x <= 9e+107)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (z / y);
	t_1 = 1.0 + (4.0 * (x / y));
	tmp = 0.0;
	if (x <= -13600000.0)
		tmp = t_1;
	elseif (x <= -8e-57)
		tmp = t_0;
	elseif (x <= 4.4e-120)
		tmp = 4.0;
	elseif (x <= 2.6e-57)
		tmp = t_0;
	elseif (x <= 1.16e+17)
		tmp = 4.0;
	elseif (x <= 9e+107)
		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[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -13600000.0], t$95$1, If[LessEqual[x, -8e-57], t$95$0, If[LessEqual[x, 4.4e-120], 4.0, If[LessEqual[x, 2.6e-57], t$95$0, If[LessEqual[x, 1.16e+17], 4.0, If[LessEqual[x, 9e+107], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.36e7 or 9e107 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 77.9%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{x}{y}} \]

   if -1.36e7 < x < -7.99999999999999964e-57 or 4.40000000000000025e-120 < x < 2.59999999999999985e-57 or 1.16e17 < x < 9e107

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 98.0%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in z around inf 60.5%

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

      1. *-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -7.99999999999999964e-57 < x < 4.40000000000000025e-120 or 2.59999999999999985e-57 < x < 1.16e17

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 55.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13600000:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-57}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-120}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+107}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 56.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -4 \cdot \frac{z}{y}\\ t_1 := 1 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -235000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-303}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -4.0 (/ z y)))) (t_1 (+ 1.0 (* 4.0 (/ x y)))))
   (if (<= x -235000000.0)
     t_1
     (if (<= x -6.8e-138)
       t_0
       (if (<= x -1.45e-303)
         4.0
         (if (<= x 6.2e-57)
           t_0
           (if (<= x 4.2e+17) 4.0 (if (<= x 1.4e+109) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (-4.0 * (z / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -235000000.0) {
		tmp = t_1;
	} else if (x <= -6.8e-138) {
		tmp = t_0;
	} else if (x <= -1.45e-303) {
		tmp = 4.0;
	} else if (x <= 6.2e-57) {
		tmp = t_0;
	} else if (x <= 4.2e+17) {
		tmp = 4.0;
	} else if (x <= 1.4e+109) {
		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 = 1.0d0 + ((-4.0d0) * (z / y))
    t_1 = 1.0d0 + (4.0d0 * (x / y))
    if (x <= (-235000000.0d0)) then
        tmp = t_1
    else if (x <= (-6.8d-138)) then
        tmp = t_0
    else if (x <= (-1.45d-303)) then
        tmp = 4.0d0
    else if (x <= 6.2d-57) then
        tmp = t_0
    else if (x <= 4.2d+17) then
        tmp = 4.0d0
    else if (x <= 1.4d+109) 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 = 1.0 + (-4.0 * (z / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -235000000.0) {
		tmp = t_1;
	} else if (x <= -6.8e-138) {
		tmp = t_0;
	} else if (x <= -1.45e-303) {
		tmp = 4.0;
	} else if (x <= 6.2e-57) {
		tmp = t_0;
	} else if (x <= 4.2e+17) {
		tmp = 4.0;
	} else if (x <= 1.4e+109) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (-4.0 * (z / y))
	t_1 = 1.0 + (4.0 * (x / y))
	tmp = 0
	if x <= -235000000.0:
		tmp = t_1
	elif x <= -6.8e-138:
		tmp = t_0
	elif x <= -1.45e-303:
		tmp = 4.0
	elif x <= 6.2e-57:
		tmp = t_0
	elif x <= 4.2e+17:
		tmp = 4.0
	elif x <= 1.4e+109:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(-4.0 * Float64(z / y)))
	t_1 = Float64(1.0 + Float64(4.0 * Float64(x / y)))
	tmp = 0.0
	if (x <= -235000000.0)
		tmp = t_1;
	elseif (x <= -6.8e-138)
		tmp = t_0;
	elseif (x <= -1.45e-303)
		tmp = 4.0;
	elseif (x <= 6.2e-57)
		tmp = t_0;
	elseif (x <= 4.2e+17)
		tmp = 4.0;
	elseif (x <= 1.4e+109)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (-4.0 * (z / y));
	t_1 = 1.0 + (4.0 * (x / y));
	tmp = 0.0;
	if (x <= -235000000.0)
		tmp = t_1;
	elseif (x <= -6.8e-138)
		tmp = t_0;
	elseif (x <= -1.45e-303)
		tmp = 4.0;
	elseif (x <= 6.2e-57)
		tmp = t_0;
	elseif (x <= 4.2e+17)
		tmp = 4.0;
	elseif (x <= 1.4e+109)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -235000000.0], t$95$1, If[LessEqual[x, -6.8e-138], t$95$0, If[LessEqual[x, -1.45e-303], 4.0, If[LessEqual[x, 6.2e-57], t$95$0, If[LessEqual[x, 4.2e+17], 4.0, If[LessEqual[x, 1.4e+109], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.35e8 or 1.4000000000000001e109 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 77.9%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{x}{y}} \]

   if -2.35e8 < x < -6.8000000000000003e-138 or -1.45000000000000007e-303 < x < 6.19999999999999952e-57 or 4.2e17 < x < 1.4000000000000001e109

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in z around inf 58.1%

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

      1. *-commutative 58.1%

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

   6. Simplified 58.1%

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

   if -6.8000000000000003e-138 < x < -1.45000000000000007e-303 or 6.19999999999999952e-57 < x < 4.2e17

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 62.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -235000000:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-138}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-303}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+109}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;x \leq -15500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-127}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (* -4.0 (/ z y))))
   (if (<= x -15500000.0)
     t_0
     (if (<= x -2.6e-58)
       t_1
       (if (<= x 1.9e-127)
         4.0
         (if (<= x 1.2e-55)
           t_1
           (if (<= x 2.05e+20) 4.0 (if (<= x 7.2e+110) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -15500000.0) {
		tmp = t_0;
	} else if (x <= -2.6e-58) {
		tmp = t_1;
	} else if (x <= 1.9e-127) {
		tmp = 4.0;
	} else if (x <= 1.2e-55) {
		tmp = t_1;
	} else if (x <= 2.05e+20) {
		tmp = 4.0;
	} else if (x <= 7.2e+110) {
		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 / y)
    t_1 = (-4.0d0) * (z / y)
    if (x <= (-15500000.0d0)) then
        tmp = t_0
    else if (x <= (-2.6d-58)) then
        tmp = t_1
    else if (x <= 1.9d-127) then
        tmp = 4.0d0
    else if (x <= 1.2d-55) then
        tmp = t_1
    else if (x <= 2.05d+20) then
        tmp = 4.0d0
    else if (x <= 7.2d+110) 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 / y);
	double t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -15500000.0) {
		tmp = t_0;
	} else if (x <= -2.6e-58) {
		tmp = t_1;
	} else if (x <= 1.9e-127) {
		tmp = 4.0;
	} else if (x <= 1.2e-55) {
		tmp = t_1;
	} else if (x <= 2.05e+20) {
		tmp = 4.0;
	} else if (x <= 7.2e+110) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = -4.0 * (z / y)
	tmp = 0
	if x <= -15500000.0:
		tmp = t_0
	elif x <= -2.6e-58:
		tmp = t_1
	elif x <= 1.9e-127:
		tmp = 4.0
	elif x <= 1.2e-55:
		tmp = t_1
	elif x <= 2.05e+20:
		tmp = 4.0
	elif x <= 7.2e+110:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(-4.0 * Float64(z / y))
	tmp = 0.0
	if (x <= -15500000.0)
		tmp = t_0;
	elseif (x <= -2.6e-58)
		tmp = t_1;
	elseif (x <= 1.9e-127)
		tmp = 4.0;
	elseif (x <= 1.2e-55)
		tmp = t_1;
	elseif (x <= 2.05e+20)
		tmp = 4.0;
	elseif (x <= 7.2e+110)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = -4.0 * (z / y);
	tmp = 0.0;
	if (x <= -15500000.0)
		tmp = t_0;
	elseif (x <= -2.6e-58)
		tmp = t_1;
	elseif (x <= 1.9e-127)
		tmp = 4.0;
	elseif (x <= 1.2e-55)
		tmp = t_1;
	elseif (x <= 2.05e+20)
		tmp = 4.0;
	elseif (x <= 7.2e+110)
		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 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -15500000.0], t$95$0, If[LessEqual[x, -2.6e-58], t$95$1, If[LessEqual[x, 1.9e-127], 4.0, If[LessEqual[x, 1.2e-55], t$95$1, If[LessEqual[x, 2.05e+20], 4.0, If[LessEqual[x, 7.2e+110], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e7 or 7.1999999999999994e110 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 95.9%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 95.9%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 95.9%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 95.9%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   9. Simplified 76.2%

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

   if -1.55e7 < x < -2.60000000000000007e-58 or 1.90000000000000001e-127 < x < 1.19999999999999996e-55 or 2.05e20 < x < 7.1999999999999994e110

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 98.0%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in z around inf 60.5%

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

      1. *-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -2.60000000000000007e-58 < x < 1.90000000000000001e-127 or 1.19999999999999996e-55 < x < 2.05e20

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 55.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15500000:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-127}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-55}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.75e-40)
   (+ 4.0 (/ x (* y 0.25)))
   (if (<= y 2.1e+40) (+ 1.0 (/ 4.0 (/ y (- x z)))) (+ 4.0 (* -4.0 (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-40) {
		tmp = 4.0 + (x / (y * 0.25));
	} else if (y <= 2.1e+40) {
		tmp = 1.0 + (4.0 / (y / (x - z)));
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.75d-40)) then
        tmp = 4.0d0 + (x / (y * 0.25d0))
    else if (y <= 2.1d+40) then
        tmp = 1.0d0 + (4.0d0 / (y / (x - z)))
    else
        tmp = 4.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-40) {
		tmp = 4.0 + (x / (y * 0.25));
	} else if (y <= 2.1e+40) {
		tmp = 1.0 + (4.0 / (y / (x - z)));
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.75e-40:
		tmp = 4.0 + (x / (y * 0.25))
	elif y <= 2.1e+40:
		tmp = 1.0 + (4.0 / (y / (x - z)))
	else:
		tmp = 4.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.75e-40)
		tmp = Float64(4.0 + Float64(x / Float64(y * 0.25)));
	elseif (y <= 2.1e+40)
		tmp = Float64(1.0 + Float64(4.0 / Float64(y / Float64(x - z))));
	else
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.75e-40)
		tmp = 4.0 + (x / (y * 0.25));
	elseif (y <= 2.1e+40)
		tmp = 1.0 + (4.0 / (y / (x - z)));
	else
		tmp = 4.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.75e-40], N[(4.0 + N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+40], N[(1.0 + N[(4.0 / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7500000000000001e-40

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. /-rgt-identity 86.7%

         \[\leadsto 4 + \frac{\color{blue}{\frac{4 \cdot x}{1}}}{y} \]

      3. *-commutative 86.7%

         \[\leadsto 4 + \frac{\frac{\color{blue}{x \cdot 4}}{1}}{y} \]

      4. associate-/l* 86.7%

         \[\leadsto 4 + \frac{\color{blue}{\frac{x}{\frac{1}{4}}}}{y} \]

      5. metadata-eval 86.7%

         \[\leadsto 4 + \frac{\frac{x}{\color{blue}{0.25}}}{y} \]

      6. associate-/r* 86.7%

         \[\leadsto 4 + \color{blue}{\frac{x}{0.25 \cdot y}} \]

      7. *-commutative 86.7%

         \[\leadsto 4 + \frac{x}{\color{blue}{y \cdot 0.25}} \]

   9. Simplified 86.7%

      \[\leadsto 4 + \color{blue}{\frac{x}{y \cdot 0.25}} \]

   if -1.7500000000000001e-40 < y < 2.1000000000000001e40

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 94.4%

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

      1. associate-*r/ 94.4%

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

      2. associate-/l* 94.2%

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

   6. Simplified 94.2%

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

   if 2.1000000000000001e40 < y

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 86.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 6: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+128} \lor \neg \left(x \leq 9.5 \cdot 10^{+108}\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.7e+128) (not (<= x 9.5e+108)))
   (+ 1.0 (* 4.0 (/ x y)))
   (+ 4.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.7e+128) || !(x <= 9.5e+108)) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-5.7d+128)) .or. (.not. (x <= 9.5d+108))) then
        tmp = 1.0d0 + (4.0d0 * (x / y))
    else
        tmp = 4.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.7e+128) || !(x <= 9.5e+108)) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.7e+128) or not (x <= 9.5e+108):
		tmp = 1.0 + (4.0 * (x / y))
	else:
		tmp = 4.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.7e+128) || !(x <= 9.5e+108))
		tmp = Float64(1.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.7e+128) || ~((x <= 9.5e+108)))
		tmp = 1.0 + (4.0 * (x / y));
	else
		tmp = 4.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.7e+128], N[Not[LessEqual[x, 9.5e+108]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.70000000000000024e128 or 9.50000000000000097e108 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 88.0%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{x}{y}} \]

   if -5.70000000000000024e128 < x < 9.50000000000000097e108

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 82.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+128} \lor \neg \left(x \leq 9.5 \cdot 10^{+108}\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 7: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+24} \lor \neg \left(z \leq 1.25 \cdot 10^{+121}\right):\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.1e+24) (not (<= z 1.25e+121)))
   (+ 4.0 (* -4.0 (/ z y)))
   (+ 4.0 (/ x (* y 0.25)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.1e+24) || !(z <= 1.25e+121)) {
		tmp = 4.0 + (-4.0 * (z / y));
	} else {
		tmp = 4.0 + (x / (y * 0.25));
	}
	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 <= (-4.1d+24)) .or. (.not. (z <= 1.25d+121))) then
        tmp = 4.0d0 + ((-4.0d0) * (z / y))
    else
        tmp = 4.0d0 + (x / (y * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.1e+24) || !(z <= 1.25e+121)) {
		tmp = 4.0 + (-4.0 * (z / y));
	} else {
		tmp = 4.0 + (x / (y * 0.25));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.1e+24) or not (z <= 1.25e+121):
		tmp = 4.0 + (-4.0 * (z / y))
	else:
		tmp = 4.0 + (x / (y * 0.25))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.1e+24) || !(z <= 1.25e+121))
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	else
		tmp = Float64(4.0 + Float64(x / Float64(y * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.1e+24) || ~((z <= 1.25e+121)))
		tmp = 4.0 + (-4.0 * (z / y));
	else
		tmp = 4.0 + (x / (y * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.1e+24], N[Not[LessEqual[z, 1.25e+121]], $MachinePrecision]], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000001e24 or 1.25000000000000002e121 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 88.0%

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

   if -4.1000000000000001e24 < z < 1.25000000000000002e121

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 90.2%

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

      1. associate-*r/ 90.2%

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

      2. /-rgt-identity 90.2%

         \[\leadsto 4 + \frac{\color{blue}{\frac{4 \cdot x}{1}}}{y} \]

      3. *-commutative 90.2%

         \[\leadsto 4 + \frac{\frac{\color{blue}{x \cdot 4}}{1}}{y} \]

      4. associate-/l* 90.2%

         \[\leadsto 4 + \frac{\color{blue}{\frac{x}{\frac{1}{4}}}}{y} \]

      5. metadata-eval 90.2%

         \[\leadsto 4 + \frac{\frac{x}{\color{blue}{0.25}}}{y} \]

      6. associate-/r* 90.2%

         \[\leadsto 4 + \color{blue}{\frac{x}{0.25 \cdot y}} \]

      7. *-commutative 90.2%

         \[\leadsto 4 + \frac{x}{\color{blue}{y \cdot 0.25}} \]

   9. Simplified 90.2%

      \[\leadsto 4 + \color{blue}{\frac{x}{y \cdot 0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+24} \lor \neg \left(z \leq 1.25 \cdot 10^{+121}\right):\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \end{array} \]

Alternative 8: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+40}:\\ \;\;\;\;\frac{4 \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.75e-40)
   (+ 4.0 (/ x (* y 0.25)))
   (if (<= y 1.15e+40) (/ (* 4.0 (- x z)) y) (+ 4.0 (* -4.0 (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-40) {
		tmp = 4.0 + (x / (y * 0.25));
	} else if (y <= 1.15e+40) {
		tmp = (4.0 * (x - z)) / y;
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.75d-40)) then
        tmp = 4.0d0 + (x / (y * 0.25d0))
    else if (y <= 1.15d+40) then
        tmp = (4.0d0 * (x - z)) / y
    else
        tmp = 4.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-40) {
		tmp = 4.0 + (x / (y * 0.25));
	} else if (y <= 1.15e+40) {
		tmp = (4.0 * (x - z)) / y;
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.75e-40:
		tmp = 4.0 + (x / (y * 0.25))
	elif y <= 1.15e+40:
		tmp = (4.0 * (x - z)) / y
	else:
		tmp = 4.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.75e-40)
		tmp = Float64(4.0 + Float64(x / Float64(y * 0.25)));
	elseif (y <= 1.15e+40)
		tmp = Float64(Float64(4.0 * Float64(x - z)) / y);
	else
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.75e-40)
		tmp = 4.0 + (x / (y * 0.25));
	elseif (y <= 1.15e+40)
		tmp = (4.0 * (x - z)) / y;
	else
		tmp = 4.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.75e-40], N[(4.0 + N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+40], N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7500000000000001e-40

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. /-rgt-identity 86.7%

         \[\leadsto 4 + \frac{\color{blue}{\frac{4 \cdot x}{1}}}{y} \]

      3. *-commutative 86.7%

         \[\leadsto 4 + \frac{\frac{\color{blue}{x \cdot 4}}{1}}{y} \]

      4. associate-/l* 86.7%

         \[\leadsto 4 + \frac{\color{blue}{\frac{x}{\frac{1}{4}}}}{y} \]

      5. metadata-eval 86.7%

         \[\leadsto 4 + \frac{\frac{x}{\color{blue}{0.25}}}{y} \]

      6. associate-/r* 86.7%

         \[\leadsto 4 + \color{blue}{\frac{x}{0.25 \cdot y}} \]

      7. *-commutative 86.7%

         \[\leadsto 4 + \frac{x}{\color{blue}{y \cdot 0.25}} \]

   9. Simplified 86.7%

      \[\leadsto 4 + \color{blue}{\frac{x}{y \cdot 0.25}} \]

   if -1.7500000000000001e-40 < y < 1.14999999999999997e40

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 96.2%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.2%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 96.2%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 96.2%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in y around 0 93.7%

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

      1. associate-*r/ 93.7%

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

   9. Simplified 93.7%

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

   if 1.14999999999999997e40 < y

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 86.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;4 + \frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+40}:\\ \;\;\;\;\frac{4 \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 9: 53.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-22} \lor \neg \left(x \leq 4.1 \cdot 10^{+57}\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.7e-22) (not (<= x 4.1e+57))) (* x (/ 4.0 y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-22) || !(x <= 4.1e+57)) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	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 <= (-1.7d-22)) .or. (.not. (x <= 4.1d+57))) then
        tmp = x * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-22) || !(x <= 4.1e+57)) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.7e-22) or not (x <= 4.1e+57):
		tmp = x * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.7e-22) || !(x <= 4.1e+57))
		tmp = Float64(x * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.7e-22) || ~((x <= 4.1e+57)))
		tmp = x * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.7e-22], N[Not[LessEqual[x, 4.1e+57]], $MachinePrecision]], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-22 or 4.1e57 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 95.7%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 95.7%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 95.7%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in x around inf 69.1%

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

      1. *-commutative 69.1%

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

   9. Simplified 69.1%

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

   10. Taylor expanded in x around 0 69.1%

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

       1. associate-*r/ 69.1%

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

       2. associate-*l/ 69.0%

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

       3. *-commutative 69.0%

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

   12. Simplified 69.0%

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

   if -1.6999999999999999e-22 < x < 4.1e57

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 49.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-22} \lor \neg \left(x \leq 4.1 \cdot 10^{+57}\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 10: 53.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-22} \lor \neg \left(x \leq 8 \cdot 10^{+58}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e-22) (not (<= x 8e+58))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-22) || !(x <= 8e+58)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	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-22)) .or. (.not. (x <= 8d+58))) then
        tmp = 4.0d0 * (x / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-22) || !(x <= 8e+58)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e-22) or not (x <= 8e+58):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e-22) || !(x <= 8e+58))
		tmp = Float64(4.0 * Float64(x / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e-22) || ~((x <= 8e+58)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e-22], N[Not[LessEqual[x, 8e+58]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-22 or 7.99999999999999955e58 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 95.7%

      \[\leadsto \color{blue}{1 + \left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{\left(4 \cdot \left(0.75 - \frac{z}{y}\right) + 4 \cdot \frac{x}{y}\right) + 1} \]

      2. distribute-lft-out 95.7%

         \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right)} + 1 \]

   6. Simplified 95.7%

      \[\leadsto \color{blue}{4 \cdot \left(\left(0.75 - \frac{z}{y}\right) + \frac{x}{y}\right) + 1} \]

   7. Taylor expanded in x around inf 69.1%

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

      1. *-commutative 69.1%

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

   9. Simplified 69.1%

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

   if -2.0000000000000001e-22 < x < 7.99999999999999955e58

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 49.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-22} \lor \neg \left(x \leq 8 \cdot 10^{+58}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 11: 34.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

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
end function

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.8%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. +-commutative 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   3. fma-def 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

4. Taylor expanded in y around inf 33.1%

   \[\leadsto \color{blue}{4} \]

5. Final simplification 33.1%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))