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

Percentage Accurate: 99.8% → 100.0%

Time: 10.5s

Alternatives: 10

Speedup: 1.5×

• 21.0% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4, \frac{x - z}{y}, 4\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := \frac{z}{y \cdot -0.25}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (/ z (* y -0.25))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e+16)
       t_2
       (if (<= t_1 5.0)
         4.0
         (if (<= t_1 5e+95) (* z (/ -4.0 y)) (if (<= t_1 5e+267) t_0 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = z / (y * -0.25);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+95) {
		tmp = z * (-4.0 / y);
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = z / (y * -0.25);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+95) {
		tmp = z * (-4.0 / y);
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = z / (y * -0.25)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= -1e+16:
		tmp = t_2
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 5e+95:
		tmp = z * (-4.0 / y)
	elif t_1 <= 5e+267:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(z / Float64(y * -0.25))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+95)
		tmp = Float64(z * Float64(-4.0 / y));
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = z / (y * -0.25);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+95)
		tmp = z * (-4.0 / y);
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	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[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(z / N[(y * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e+16], t$95$2, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 5e+95], N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -inf.0 or 5.00000000000000025e95 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 4.9999999999999999e267

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   if -inf.0 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e16 or 4.9999999999999999e267 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 66.0

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

   7. Applied rewrites 66.0%

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

   9. Applied rewrites 66.1%

      \[\leadsto \frac{z}{\color{blue}{y \cdot -0.25}} \]

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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

      \[\leadsto \color{blue}{4} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.9%

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

   if 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5.00000000000000025e95

   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 z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -\infty:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{y \cdot -0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := z \cdot \frac{-4}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (* z (/ -4.0 y))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e+16)
       t_2
       (if (<= t_1 5.0)
         4.0
         (if (<= t_1 5e+95) t_2 (if (<= t_1 5e+267) t_0 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = z * (-4.0 / y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = z * (-4.0 / y);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = z * (-4.0 / y)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= -1e+16:
		tmp = t_2
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 5e+95:
		tmp = t_2
	elif t_1 <= 5e+267:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(z * Float64(-4.0 / y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = z * (-4.0 / y);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	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[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e+16], t$95$2, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 5e+95], t$95$2, If[LessEqual[t$95$1, 5e+267], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -inf.0 or 5.00000000000000025e95 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 4.9999999999999999e267

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   if -inf.0 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e16 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5.00000000000000025e95 or 4.9999999999999999e267 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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

      \[\leadsto \color{blue}{4} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -\infty:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{4 \cdot \left(x - z\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_0 -2e+19)
     (/ (* 4.0 (- x z)) y)
     (if (<= t_0 50000000000.0)
       (fma -4.0 (/ z y) 4.0)
       (* (- x z) (/ 4.0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_0 <= -2e+19) {
		tmp = (4.0 * (x - z)) / y;
	} else if (t_0 <= 50000000000.0) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = (x - z) * (4.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_0 <= -2e+19)
		tmp = Float64(Float64(4.0 * Float64(x - z)) / y);
	elseif (t_0 <= 50000000000.0)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+19], N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 50000000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -2e19 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e10

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. associate-+r+ N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   12. Step-by-step derivation

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r/ N/A

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

       5. *-inverses N/A

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

       6. metadata-eval N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. distribute-lft-neg-in N/A

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

       9. metadata-eval N/A

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

       10. associate-+r+ N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. +-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

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

   if 5e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 97.0

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

   10. Applied rewrites 97.0%

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

   12. Applied rewrites 99.1%

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

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - z\right)}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x z)) y)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -2e+19)
     t_0
     (if (<= t_1 50000000000.0) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - z)) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = t_0;
	} else if (t_1 <= 50000000000.0) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - z)) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = t_0;
	elseif (t_1 <= 50000000000.0)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], t$95$0, If[LessEqual[t$95$1, 50000000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e19 or 5e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if -2e19 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e10

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. associate-+r+ N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   12. Step-by-step derivation

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r/ N/A

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

       5. *-inverses N/A

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

       6. metadata-eval N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. distribute-lft-neg-in N/A

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

       9. metadata-eval N/A

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

       10. associate-+r+ N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. +-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -1e+16) t_0 (if (<= t_1 5.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -1e+16) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} 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 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-1d+16)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else
        tmp = t_0
    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 * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -1e+16) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -1e+16:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -1e+16)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -1e+16)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	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[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+16], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e16 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 54.8

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

   5. Applied rewrites 54.8%

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

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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

      \[\leadsto \color{blue}{4} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.9%

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

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 4.0)))
   (if (<= z -6.5e-27) t_0 (if (<= z 0.16) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-4.0, (z / y), 4.0);
	double tmp;
	if (z <= -6.5e-27) {
		tmp = t_0;
	} else if (z <= 0.16) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-4.0, Float64(z / y), 4.0)
	tmp = 0.0
	if (z <= -6.5e-27)
		tmp = t_0;
	elseif (z <= 0.16)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision]}, If[LessEqual[z, -6.5e-27], t$95$0, If[LessEqual[z, 0.16], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000025e-27 or 0.160000000000000003 < z

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. associate-+r+ N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   12. Step-by-step derivation

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r/ N/A

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

       5. *-inverses N/A

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

       6. metadata-eval N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. distribute-lft-neg-in N/A

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

       9. metadata-eval N/A

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

       10. associate-+r+ N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. +-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-/.f64 88.0

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

   13. Applied rewrites 88.0%

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

   if -6.50000000000000025e-27 < z < 0.160000000000000003

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. neg-sub0 N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(0 - \left(-1 \cdot \frac{x}{y} - \frac{3}{4}\right)\right)} \]

      17. sub-neg N/A

          \[\leadsto 1 + 4 \cdot \left(0 - \color{blue}{\left(-1 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto 1 + 4 \cdot \left(0 - \left(-1 \cdot \frac{x}{y} + \color{blue}{\frac{-3}{4}}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto 1 + 4 \cdot \left(0 - \color{blue}{\left(\frac{-3}{4} + -1 \cdot \frac{x}{y}\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto 1 + 4 \cdot \left(0 - \left(\frac{-3}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

   7. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -1.1e+141) t_0 (if (<= x 6.5e+141) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -1.1e+141) {
		tmp = t_0;
	} else if (x <= 6.5e+141) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -1.1e+141)
		tmp = t_0;
	elseif (x <= 6.5e+141)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+141], t$95$0, If[LessEqual[x, 6.5e+141], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e141 or 6.50000000000000053e141 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 81.0%

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

   if -1.1e141 < x < 6.50000000000000053e141

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. associate-+r+ N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 90.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   12. Step-by-step derivation

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r/ N/A

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

       5. *-inverses N/A

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

       6. metadata-eval N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. distribute-lft-neg-in N/A

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

       9. metadata-eval N/A

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

       10. associate-+r+ N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. +-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-/.f64 90.8

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

   13. Applied rewrites 90.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -1.1e+141) t_0 (if (<= x 6.5e+141) (fma z (/ -4.0 y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -1.1e+141) {
		tmp = t_0;
	} else if (x <= 6.5e+141) {
		tmp = fma(z, (-4.0 / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -1.1e+141)
		tmp = t_0;
	elseif (x <= 6.5e+141)
		tmp = fma(z, Float64(-4.0 / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+141], t$95$0, If[LessEqual[x, 6.5e+141], N[(z * N[(-4.0 / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e141 or 6.50000000000000053e141 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 81.0%

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

   if -1.1e141 < x < 6.50000000000000053e141

   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 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. associate-+l+ N/A

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

   5. Applied rewrites 90.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.4% accurate, 31.0× 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 99.1%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{4} \]
4. Step-by-step derivation

5. Applied rewrites 35.9%

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

Reproduce

herbie shell --seed 2024219 
(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)))