Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Percentage Accurate: 95.6% → 99.2%

Time: 10.8s

Alternatives: 10

Speedup: 1.0×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -2 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot 3 \leq 8 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))))
   (if (<= (* z 3.0) -2e+32)
     t_1
     (if (<= (* z 3.0) 8e-96)
       (fma 0.3333333333333333 (/ (- (/ t y) y) z) x)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	double tmp;
	if ((z * 3.0) <= -2e+32) {
		tmp = t_1;
	} else if ((z * 3.0) <= 8e-96) {
		tmp = fma(0.3333333333333333, (((t / y) - y) / z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -2e+32)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 8e-96)
		tmp = fma(0.3333333333333333, Float64(Float64(Float64(t / y) - y) / z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -2e+32], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 8e-96], N[(0.3333333333333333 * N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -2.00000000000000011e32 or 7.9999999999999993e-96 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   if -2.00000000000000011e32 < (*.f64 z #s(literal 3 binary64)) < 7.9999999999999993e-96

   1. Initial program 89.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, t \cdot \frac{0.3333333333333333}{z}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma 0.3333333333333333 (/ (- (/ t y) y) z) x)))
   (if (<= y -2.2e-64)
     t_1
     (if (<= y 8.5e-107) (/ (fma y x (* t (/ 0.3333333333333333 z))) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(0.3333333333333333, (((t / y) - y) / z), x);
	double tmp;
	if (y <= -2.2e-64) {
		tmp = t_1;
	} else if (y <= 8.5e-107) {
		tmp = fma(y, x, (t * (0.3333333333333333 / z))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(0.3333333333333333, Float64(Float64(Float64(t / y) - y) / z), x)
	tmp = 0.0
	if (y <= -2.2e-64)
		tmp = t_1;
	elseif (y <= 8.5e-107)
		tmp = Float64(fma(y, x, Float64(t * Float64(0.3333333333333333 / z))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.2e-64], t$95$1, If[LessEqual[y, 8.5e-107], N[(N[(y * x + N[(t * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e-64 or 8.49999999999999956e-107 < y

   1. Initial program 98.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.6

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   if -2.2e-64 < y < 8.49999999999999956e-107

   1. Initial program 88.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 86.2

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 96.9

          \[\leadsto \frac{\mathsf{fma}\left(y, x, t \cdot \color{blue}{\frac{0.3333333333333333}{z}}\right)}{y} \]

   8. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, x, t \cdot \frac{0.3333333333333333}{z}\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma 0.3333333333333333 (/ (- (/ t y) y) z) x)))
   (if (<= y -1.22e-79)
     t_1
     (if (<= y 6.8e-104) (/ (fma 0.3333333333333333 (/ t z) (* x y)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(0.3333333333333333, (((t / y) - y) / z), x);
	double tmp;
	if (y <= -1.22e-79) {
		tmp = t_1;
	} else if (y <= 6.8e-104) {
		tmp = fma(0.3333333333333333, (t / z), (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(0.3333333333333333, Float64(Float64(Float64(t / y) - y) / z), x)
	tmp = 0.0
	if (y <= -1.22e-79)
		tmp = t_1;
	elseif (y <= 6.8e-104)
		tmp = Float64(fma(0.3333333333333333, Float64(t / z), Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.22e-79], t$95$1, If[LessEqual[y, 6.8e-104], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22e-79 or 6.80000000000000031e-104 < y

   1. Initial program 98.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.6

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   if -1.22e-79 < y < 6.80000000000000031e-104

   1. Initial program 88.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.8

         \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, \color{blue}{y \cdot x}\right)}{y} \]

   5. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, y \cdot x\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.18e+182)
   (fma 0.3333333333333333 (/ (- (/ t y) y) z) x)
   (fma 0.3333333333333333 (/ t (* z y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.18e+182) {
		tmp = fma(0.3333333333333333, (((t / y) - y) / z), x);
	} else {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.18e+182)
		tmp = fma(0.3333333333333333, Float64(Float64(Float64(t / y) - y) / z), x);
	else
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.18e+182], N[(0.3333333333333333 * N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1799999999999999e182

   1. Initial program 93.9%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 96.5

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   if 1.1799999999999999e182 < t

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 69.9

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\frac{t}{y \cdot z}}, x\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 91.9

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{t}{\color{blue}{y \cdot z}}, x\right) \]

   8. Simplified 91.9%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{t}{y \cdot z}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -25000000000000.0)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 4.2e-5)
     (fma 0.3333333333333333 (/ t (* z y)) x)
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -25000000000000.0) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 4.2e-5) {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -25000000000000.0)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 4.2e-5)
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -25000000000000.0], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 4.2e-5], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e13

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 95.8

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-0.3333333333333333}{z}}, x\right) \]

   5. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -2.5e13 < y < 4.19999999999999977e-5

   1. Initial program 90.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 88.9

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\frac{t}{y \cdot z}}, x\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 87.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{t}{\color{blue}{y \cdot z}}, x\right) \]

   8. Simplified 87.8%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{t}{y \cdot z}}, x\right) \]

   if 4.19999999999999977e-5 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{-1 \cdot \frac{y}{z}}, x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.5

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{-y}}{z}, x\right) \]

   8. Simplified 94.5%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{-y}{z}}, x\right) \]

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 94.5

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

   11. Simplified 94.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-32}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.16e-7)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 1.45e-32)
     (/ (* t 0.3333333333333333) (* z y))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.16e-7) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 1.45e-32) {
		tmp = (t * 0.3333333333333333) / (z * y);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.16e-7)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 1.45e-32)
		tmp = Float64(Float64(t * 0.3333333333333333) / Float64(z * y));
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.16e-7], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.45e-32], N[(N[(t * 0.3333333333333333), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1600000000000001e-7

   1. Initial program 98.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 94.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-0.3333333333333333}{z}}, x\right) \]

   5. Simplified 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]

   if -1.1600000000000001e-7 < y < 1.44999999999999998e-32

   1. Initial program 89.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot t}}{y \cdot z} \]

      4. lower-*.f64 59.3

         \[\leadsto \frac{0.3333333333333333 \cdot t}{\color{blue}{y \cdot z}} \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot t}{y \cdot z}} \]

   if 1.44999999999999998e-32 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{-1 \cdot \frac{y}{z}}, x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.5

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{-y}}{z}, x\right) \]

   8. Simplified 90.5%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{-y}{z}}, x\right) \]

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 90.5

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

   11. Simplified 90.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-32}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.15e+77)
   (/ (* y -0.3333333333333333) z)
   (if (<= y 6.6e-8) (/ (* x y) y) (* -0.3333333333333333 (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.15e+77) {
		tmp = (y * -0.3333333333333333) / z;
	} else if (y <= 6.6e-8) {
		tmp = (x * y) / y;
	} else {
		tmp = -0.3333333333333333 * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.15d+77)) then
        tmp = (y * (-0.3333333333333333d0)) / z
    else if (y <= 6.6d-8) then
        tmp = (x * y) / y
    else
        tmp = (-0.3333333333333333d0) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.15e+77) {
		tmp = (y * -0.3333333333333333) / z;
	} else if (y <= 6.6e-8) {
		tmp = (x * y) / y;
	} else {
		tmp = -0.3333333333333333 * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.15e+77:
		tmp = (y * -0.3333333333333333) / z
	elif y <= 6.6e-8:
		tmp = (x * y) / y
	else:
		tmp = -0.3333333333333333 * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.15e+77)
		tmp = Float64(Float64(y * -0.3333333333333333) / z);
	elseif (y <= 6.6e-8)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.15e+77)
		tmp = (y * -0.3333333333333333) / z;
	elseif (y <= 6.6e-8)
		tmp = (x * y) / y;
	else
		tmp = -0.3333333333333333 * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.15e+77], N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 6.6e-8], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.14999999999999996e77

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-lft-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 80.7

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{\frac{t}{y}} - y\right)}{z} \]

   5. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.7

         \[\leadsto \frac{\color{blue}{y \cdot -0.3333333333333333}}{z} \]

   8. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{y \cdot -0.3333333333333333}}{z} \]

   if -2.14999999999999996e77 < y < 6.59999999999999954e-8

   1. Initial program 90.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 89.5

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 89.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 93.9

          \[\leadsto \frac{\mathsf{fma}\left(y, x, t \cdot \color{blue}{\frac{0.3333333333333333}{z}}\right)}{y} \]

   8. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, x, t \cdot \frac{0.3333333333333333}{z}\right)}{y}} \]

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 34.1

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

   11. Simplified 34.1%

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

   if 6.59999999999999954e-8 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 71.6

          \[\leadsto \left(\frac{t}{y} - y\right) \cdot \color{blue}{\frac{0.3333333333333333}{z}} \]

   8. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\frac{t}{y} - y\right) \cdot \frac{0.3333333333333333}{z}} \]

   9. Taylor expanded in t around 0

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

   11. Simplified 58.6%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y \cdot y - t}{y \cdot z}} \]

   12. Taylor expanded in y around inf

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

       1. lower-/.f64 66.4

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]

   14. Simplified 66.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.15e+77)
   (* y (/ -0.3333333333333333 z))
   (if (<= y 6.6e-8) (/ (* x y) y) (* -0.3333333333333333 (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.15e+77) {
		tmp = y * (-0.3333333333333333 / z);
	} else if (y <= 6.6e-8) {
		tmp = (x * y) / y;
	} else {
		tmp = -0.3333333333333333 * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.15d+77)) then
        tmp = y * ((-0.3333333333333333d0) / z)
    else if (y <= 6.6d-8) then
        tmp = (x * y) / y
    else
        tmp = (-0.3333333333333333d0) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.15e+77) {
		tmp = y * (-0.3333333333333333 / z);
	} else if (y <= 6.6e-8) {
		tmp = (x * y) / y;
	} else {
		tmp = -0.3333333333333333 * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.15e+77:
		tmp = y * (-0.3333333333333333 / z)
	elif y <= 6.6e-8:
		tmp = (x * y) / y
	else:
		tmp = -0.3333333333333333 * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.15e+77)
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	elseif (y <= 6.6e-8)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.15e+77)
		tmp = y * (-0.3333333333333333 / z);
	elseif (y <= 6.6e-8)
		tmp = (x * y) / y;
	else
		tmp = -0.3333333333333333 * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.15e+77], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e-8], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.14999999999999996e77

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 77.7

          \[\leadsto y \cdot \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 77.7%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]

   if -2.14999999999999996e77 < y < 6.59999999999999954e-8

   1. Initial program 90.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 89.5

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 89.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 93.9

          \[\leadsto \frac{\mathsf{fma}\left(y, x, t \cdot \color{blue}{\frac{0.3333333333333333}{z}}\right)}{y} \]

   8. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, x, t \cdot \frac{0.3333333333333333}{z}\right)}{y}} \]

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 34.1

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

   11. Simplified 34.1%

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

   if 6.59999999999999954e-8 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 71.6

          \[\leadsto \left(\frac{t}{y} - y\right) \cdot \color{blue}{\frac{0.3333333333333333}{z}} \]

   8. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\frac{t}{y} - y\right) \cdot \frac{0.3333333333333333}{z}} \]

   9. Taylor expanded in t around 0

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

   11. Simplified 58.6%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y \cdot y - t}{y \cdot z}} \]

   12. Taylor expanded in y around inf

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

       1. lower-/.f64 66.4

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]

   14. Simplified 66.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma -0.3333333333333333 (/ y z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, (y / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(y / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 93.9

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

5. Simplified 93.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{-1 \cdot \frac{y}{z}}, x\right) \]
7. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 63.0

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{-y}}{z}, x\right) \]

8. Simplified 63.0%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{-y}{z}}, x\right) \]

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 63.0

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

11. Simplified 63.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]
12. Add Preprocessing

Alternative 10: 35.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{y}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* -0.3333333333333333 (/ y z)))

C

double code(double x, double y, double z, double t) {
	return -0.3333333333333333 * (y / z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-0.3333333333333333d0) * (y / z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return -0.3333333333333333 * (y / z);
}

Python

def code(x, y, z, t):
	return -0.3333333333333333 * (y / z)

Julia

function code(x, y, z, t)
	return Float64(-0.3333333333333333 * Float64(y / z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -0.3333333333333333 * (y / z);
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 93.9

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\frac{t}{y}} - y}{z}, x\right) \]

5. Simplified 93.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y} - y}{z}, x\right)} \]

6. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 65.3

       \[\leadsto \left(\frac{t}{y} - y\right) \cdot \color{blue}{\frac{0.3333333333333333}{z}} \]

8. Simplified 65.3%

   \[\leadsto \color{blue}{\left(\frac{t}{y} - y\right) \cdot \frac{0.3333333333333333}{z}} \]

9. Taylor expanded in t around 0

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

11. Simplified 55.8%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y \cdot y - t}{y \cdot z}} \]

12. Taylor expanded in y around inf

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

    1. lower-/.f64 33.8

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]

14. Simplified 33.8%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]
15. Add Preprocessing

Developer Target 1: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))