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

Percentage Accurate: 99.8% → 100.0%

Time: 7.6s

Alternatives: 10

Speedup: 1.4×

• 20.8% 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: 99.8% 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: 100.0% accurate, 1.4× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 100.0%

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

Alternative 2: 54.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-185}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+41}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ z y))) (t_1 (* 4.0 (/ x y))))
   (if (<= x -4.1e+18)
     t_1
     (if (<= x -5.4e-185)
       4.0
       (if (<= x 1.55e-251)
         t_0
         (if (<= x 8.5e+41) 4.0 (if (<= x 3.8e+132) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (z / y);
	double t_1 = 4.0 * (x / y);
	double tmp;
	if (x <= -4.1e+18) {
		tmp = t_1;
	} else if (x <= -5.4e-185) {
		tmp = 4.0;
	} else if (x <= 1.55e-251) {
		tmp = t_0;
	} else if (x <= 8.5e+41) {
		tmp = 4.0;
	} else if (x <= 3.8e+132) {
		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 = 4.0d0 * (x / y)
    if (x <= (-4.1d+18)) then
        tmp = t_1
    else if (x <= (-5.4d-185)) then
        tmp = 4.0d0
    else if (x <= 1.55d-251) then
        tmp = t_0
    else if (x <= 8.5d+41) then
        tmp = 4.0d0
    else if (x <= 3.8d+132) 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 = 4.0 * (x / y);
	double tmp;
	if (x <= -4.1e+18) {
		tmp = t_1;
	} else if (x <= -5.4e-185) {
		tmp = 4.0;
	} else if (x <= 1.55e-251) {
		tmp = t_0;
	} else if (x <= 8.5e+41) {
		tmp = 4.0;
	} else if (x <= 3.8e+132) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (z / y)
	t_1 = 4.0 * (x / y)
	tmp = 0
	if x <= -4.1e+18:
		tmp = t_1
	elif x <= -5.4e-185:
		tmp = 4.0
	elif x <= 1.55e-251:
		tmp = t_0
	elif x <= 8.5e+41:
		tmp = 4.0
	elif x <= 3.8e+132:
		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(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -4.1e+18)
		tmp = t_1;
	elseif (x <= -5.4e-185)
		tmp = 4.0;
	elseif (x <= 1.55e-251)
		tmp = t_0;
	elseif (x <= 8.5e+41)
		tmp = 4.0;
	elseif (x <= 3.8e+132)
		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 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -4.1e+18)
		tmp = t_1;
	elseif (x <= -5.4e-185)
		tmp = 4.0;
	elseif (x <= 1.55e-251)
		tmp = t_0;
	elseif (x <= 8.5e+41)
		tmp = 4.0;
	elseif (x <= 3.8e+132)
		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[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+18], t$95$1, If[LessEqual[x, -5.4e-185], 4.0, If[LessEqual[x, 1.55e-251], t$95$0, If[LessEqual[x, 8.5e+41], 4.0, If[LessEqual[x, 3.8e+132], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1e18 or 3.80000000000000006e132 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.7%

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

   if -4.1e18 < x < -5.39999999999999976e-185 or 1.55000000000000001e-251 < x < 8.49999999999999938e41

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 54.2%

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

   if -5.39999999999999976e-185 < x < 1.55000000000000001e-251 or 8.49999999999999938e41 < x < 3.80000000000000006e132

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 70.0%

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

      1. *-commutative 70.0%

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

   5. Simplified 70.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-185}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-251}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+41}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{-184}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-243}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+43}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (* 4.0 (/ x y))))
   (if (<= x -7.5e+17)
     t_1
     (if (<= x -1.04e-184)
       4.0
       (if (<= x 1.2e-243)
         t_0
         (if (<= x 1.6e+43) 4.0 (if (<= x 7.4e+132) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = 4.0 * (x / y);
	double tmp;
	if (x <= -7.5e+17) {
		tmp = t_1;
	} else if (x <= -1.04e-184) {
		tmp = 4.0;
	} else if (x <= 1.2e-243) {
		tmp = t_0;
	} else if (x <= 1.6e+43) {
		tmp = 4.0;
	} else if (x <= 7.4e+132) {
		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 = z * ((-4.0d0) / y)
    t_1 = 4.0d0 * (x / y)
    if (x <= (-7.5d+17)) then
        tmp = t_1
    else if (x <= (-1.04d-184)) then
        tmp = 4.0d0
    else if (x <= 1.2d-243) then
        tmp = t_0
    else if (x <= 1.6d+43) then
        tmp = 4.0d0
    else if (x <= 7.4d+132) 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 = z * (-4.0 / y);
	double t_1 = 4.0 * (x / y);
	double tmp;
	if (x <= -7.5e+17) {
		tmp = t_1;
	} else if (x <= -1.04e-184) {
		tmp = 4.0;
	} else if (x <= 1.2e-243) {
		tmp = t_0;
	} else if (x <= 1.6e+43) {
		tmp = 4.0;
	} else if (x <= 7.4e+132) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = 4.0 * (x / y)
	tmp = 0
	if x <= -7.5e+17:
		tmp = t_1
	elif x <= -1.04e-184:
		tmp = 4.0
	elif x <= 1.2e-243:
		tmp = t_0
	elif x <= 1.6e+43:
		tmp = 4.0
	elif x <= 7.4e+132:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -7.5e+17)
		tmp = t_1;
	elseif (x <= -1.04e-184)
		tmp = 4.0;
	elseif (x <= 1.2e-243)
		tmp = t_0;
	elseif (x <= 1.6e+43)
		tmp = 4.0;
	elseif (x <= 7.4e+132)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -7.5e+17)
		tmp = t_1;
	elseif (x <= -1.04e-184)
		tmp = 4.0;
	elseif (x <= 1.2e-243)
		tmp = t_0;
	elseif (x <= 1.6e+43)
		tmp = 4.0;
	elseif (x <= 7.4e+132)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+17], t$95$1, If[LessEqual[x, -1.04e-184], 4.0, If[LessEqual[x, 1.2e-243], t$95$0, If[LessEqual[x, 1.6e+43], 4.0, If[LessEqual[x, 7.4e+132], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e17 or 7.40000000000000022e132 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.7%

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

   if -7.5e17 < x < -1.03999999999999994e-184 or 1.2e-243 < x < 1.60000000000000007e43

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 54.2%

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

   if -1.03999999999999994e-184 < x < 1.2e-243 or 1.60000000000000007e43 < x < 7.40000000000000022e132

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 69.7%

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

   5. Simplified 69.7%

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

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21} \lor \neg \left(z \leq 2.15 \cdot 10^{-23}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.2e+21) (not (<= z 2.15e-23)))
   (* (- x z) (/ 4.0 y))
   (+ 4.0 (* x (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2e+21) || !(z <= 2.15e-23)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (x * (4.0 / 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 ((z <= (-6.2d+21)) .or. (.not. (z <= 2.15d-23))) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0 + (x * (4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2e+21) || !(z <= 2.15e-23)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (x * (4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.2e+21) or not (z <= 2.15e-23):
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0 + (x * (4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.2e+21) || !(z <= 2.15e-23))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.2e+21) || ~((z <= 2.15e-23)))
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0 + (x * (4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.2e+21], N[Not[LessEqual[z, 2.15e-23]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2e21 or 2.15000000000000001e-23 < z

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.8%

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

      1. *-lft-identity 82.8%

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

      2. associate-*l/ 82.6%

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

      3. associate-*r* 82.6%

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

      4. associate-*r/ 82.6%

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

      5. metadata-eval 82.6%

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

   5. Simplified 82.6%

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

   if -6.2e21 < z < 2.15000000000000001e-23

   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. +-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate--r+ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. metadata-eval 93.4%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 93.5%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 93.5%

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

      5. associate-*r/ 93.5%

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

      6. *-commutative 93.5%

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

      7. associate-*r/ 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+21} \lor \neg \left(z \leq 2.15 \cdot 10^{-23}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+32}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+32)
   (* (- x z) (/ 4.0 y))
   (if (<= z 1.5e-9) (+ 4.0 (/ (* 4.0 x) y)) (+ 4.0 (/ (* z -4.0) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+32) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 1.5e-9) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / 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 (z <= (-4.5d+32)) then
        tmp = (x - z) * (4.0d0 / y)
    else if (z <= 1.5d-9) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+32) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 1.5e-9) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.5e+32:
		tmp = (x - z) * (4.0 / y)
	elif z <= 1.5e-9:
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + ((z * -4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+32)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	elseif (z <= 1.5e-9)
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.5e+32)
		tmp = (x - z) * (4.0 / y);
	elseif (z <= 1.5e-9)
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + ((z * -4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.5e+32], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e-9], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5000000000000003e32

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 87.8%

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

      1. *-lft-identity 87.8%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r* 87.5%

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

      4. associate-*r/ 87.5%

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

      5. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   if -4.5000000000000003e32 < z < 1.49999999999999999e-9

   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. +-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate--r+ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 92.8%

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

      1. distribute-lft-in 92.8%

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

      2. metadata-eval 92.8%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 92.8%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 92.8%

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

      5. associate-*r/ 92.8%

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

      6. *-commutative 92.8%

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

      7. associate-*r/ 92.6%

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

   7. Simplified 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l/ 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 1.49999999999999999e-9 < z

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in x around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. associate-*r/ 86.4%

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

   6. Simplified 86.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+32}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+32}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e+32)
   (* (- x z) (/ 4.0 y))
   (if (<= z 6.8e-10) (+ 4.0 (/ (* 4.0 x) y)) (+ 4.0 (* z (/ -4.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+32) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 6.8e-10) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + (z * (-4.0 / 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 (z <= (-3.4d+32)) then
        tmp = (x - z) * (4.0d0 / y)
    else if (z <= 6.8d-10) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + (z * ((-4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+32) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 6.8e-10) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + (z * (-4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.4e+32:
		tmp = (x - z) * (4.0 / y)
	elif z <= 6.8e-10:
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + (z * (-4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+32)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	elseif (z <= 6.8e-10)
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(z * Float64(-4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.4e+32)
		tmp = (x - z) * (4.0 / y);
	elseif (z <= 6.8e-10)
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + (z * (-4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+32], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e-10], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.39999999999999979e32

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 87.8%

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

      1. *-lft-identity 87.8%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r* 87.5%

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

      4. associate-*r/ 87.5%

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

      5. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   if -3.39999999999999979e32 < z < 6.8000000000000003e-10

   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. +-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate--r+ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 92.8%

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

      1. distribute-lft-in 92.8%

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

      2. metadata-eval 92.8%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 92.8%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 92.8%

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

      5. associate-*r/ 92.8%

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

      6. *-commutative 92.8%

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

      7. associate-*r/ 92.6%

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

   7. Simplified 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l/ 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 6.8000000000000003e-10 < z

   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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. sub-neg 86.4%

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

      2. distribute-lft-in 86.4%

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

      3. metadata-eval 86.4%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \left(-\frac{z}{y}\right)\right) \]

      4. associate-+r+ 86.4%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \left(-\frac{z}{y}\right)} \]

      5. metadata-eval 86.4%

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

      6. neg-mul-1 86.4%

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

      7. associate-*r* 86.4%

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

      8. metadata-eval 86.4%

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

      9. associate-*r/ 86.4%

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

      10. *-commutative 86.4%

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

      11. associate-/l* 86.3%

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

   7. Simplified 86.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+32}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-26}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+30)
   (* (- x z) (/ 4.0 y))
   (if (<= z 6e-26) (+ 4.0 (* x (/ 4.0 y))) (+ 4.0 (* z (/ -4.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+30) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 6e-26) {
		tmp = 4.0 + (x * (4.0 / y));
	} else {
		tmp = 4.0 + (z * (-4.0 / 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 (z <= (-2.3d+30)) then
        tmp = (x - z) * (4.0d0 / y)
    else if (z <= 6d-26) then
        tmp = 4.0d0 + (x * (4.0d0 / y))
    else
        tmp = 4.0d0 + (z * ((-4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+30) {
		tmp = (x - z) * (4.0 / y);
	} else if (z <= 6e-26) {
		tmp = 4.0 + (x * (4.0 / y));
	} else {
		tmp = 4.0 + (z * (-4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.3e+30:
		tmp = (x - z) * (4.0 / y)
	elif z <= 6e-26:
		tmp = 4.0 + (x * (4.0 / y))
	else:
		tmp = 4.0 + (z * (-4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+30)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	elseif (z <= 6e-26)
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	else
		tmp = Float64(4.0 + Float64(z * Float64(-4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.3e+30)
		tmp = (x - z) * (4.0 / y);
	elseif (z <= 6e-26)
		tmp = 4.0 + (x * (4.0 / y));
	else
		tmp = 4.0 + (z * (-4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+30], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-26], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3e30

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 87.8%

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

      1. *-lft-identity 87.8%

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

      2. associate-*l/ 87.5%

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

      3. associate-*r* 87.5%

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

      4. associate-*r/ 87.5%

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

      5. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   if -2.3e30 < z < 6.00000000000000023e-26

   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. +-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate--r+ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. metadata-eval 93.4%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 93.4%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 93.4%

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

      5. associate-*r/ 93.4%

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

      6. *-commutative 93.4%

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

      7. associate-*r/ 93.2%

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

   7. Simplified 93.2%

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

   if 6.00000000000000023e-26 < z

   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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 85.7%

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

      1. sub-neg 85.7%

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

      2. distribute-lft-in 85.7%

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

      3. metadata-eval 85.7%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \left(-\frac{z}{y}\right)\right) \]

      4. associate-+r+ 85.7%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \left(-\frac{z}{y}\right)} \]

      5. metadata-eval 85.7%

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

      6. neg-mul-1 85.7%

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

      7. associate-*r* 85.7%

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

      8. metadata-eval 85.7%

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

      9. associate-*r/ 85.7%

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

      10. *-commutative 85.7%

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

      11. associate-/l* 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-26}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.26 \cdot 10^{+80}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+150}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.26e+80) 4.0 (if (<= y 2.8e+150) (* (- x z) (/ 4.0 y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.26e+80) {
		tmp = 4.0;
	} else if (y <= 2.8e+150) {
		tmp = (x - z) * (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 (y <= (-2.26d+80)) then
        tmp = 4.0d0
    else if (y <= 2.8d+150) then
        tmp = (x - z) * (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 (y <= -2.26e+80) {
		tmp = 4.0;
	} else if (y <= 2.8e+150) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.26e+80:
		tmp = 4.0
	elif y <= 2.8e+150:
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.26e+80)
		tmp = 4.0;
	elseif (y <= 2.8e+150)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.26e+80)
		tmp = 4.0;
	elseif (y <= 2.8e+150)
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.26e+80], 4.0, If[LessEqual[y, 2.8e+150], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.26e80 or 2.80000000000000009e150 < y

   1. Initial program 99.7%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.8%

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

   if -2.26e80 < y < 2.80000000000000009e150

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 86.6%

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

      1. *-lft-identity 86.6%

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

      2. associate-*l/ 86.3%

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

      3. associate-*r* 86.3%

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

      4. associate-*r/ 86.3%

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

      5. metadata-eval 86.3%

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

   5. Simplified 86.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.26 \cdot 10^{+80}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+150}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+19} \lor \neg \left(x \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.5e+19) (not (<= x 6.6e+14))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.5e+19) || !(x <= 6.6e+14)) {
		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 <= (-3.5d+19)) .or. (.not. (x <= 6.6d+14))) 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 <= -3.5e+19) || !(x <= 6.6e+14)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.5e+19) or not (x <= 6.6e+14):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.5e+19) || !(x <= 6.6e+14))
		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 <= -3.5e+19) || ~((x <= 6.6e+14)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.5e+19], N[Not[LessEqual[x, 6.6e+14]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e19 or 6.6e14 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 61.3%

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

   if -3.5e19 < x < 6.6e14

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 46.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+19} \lor \neg \left(x \leq 6.6 \cdot 10^{+14}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.8% 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. Add Preprocessing

3. Taylor expanded in y around inf 30.9%

   \[\leadsto \color{blue}{4} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(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)))