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

Percentage Accurate: 95.3% → 97.6%

Time: 10.5s

Alternatives: 10

Speedup: 1.4×

• 28.0% 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: 95.3% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 5e-122)
   (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)
   (fma (/ y z) -0.3333333333333333 (+ x (/ t (* (* z 3.0) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-122) {
		tmp = fma((0.3333333333333333 / z), ((t / y) - y), x);
	} else {
		tmp = fma((y / z), -0.3333333333333333, (x + (t / ((z * 3.0) * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 5e-122)
		tmp = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x);
	else
		tmp = fma(Float64(y / z), -0.3333333333333333, Float64(x + Float64(t / Float64(Float64(z * 3.0) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-122], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + N[(x + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 4.9999999999999999e-122

   1. Initial program 93.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

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

   if 4.9999999999999999e-122 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{y \cdot \color{blue}{\left(z \cdot 3\right)}}\right) \]

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 5e-122)
   (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)
   (fma (/ t (* z y)) 0.3333333333333333 (fma (/ y z) -0.3333333333333333 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-122) {
		tmp = fma((0.3333333333333333 / z), ((t / y) - y), x);
	} else {
		tmp = fma((t / (z * y)), 0.3333333333333333, fma((y / z), -0.3333333333333333, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 5e-122)
		tmp = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x);
	else
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, fma(Float64(y / z), -0.3333333333333333, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-122], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 4.9999999999999999e-122

   1. Initial program 93.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

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

   if 4.9999999999999999e-122 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate-/r* N/A

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

      13. div-inv N/A

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

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

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -600000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, \frac{y}{-3}, x\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -600000.0)
   (fma (/ 1.0 z) (/ y -3.0) x)
   (if (<= y 6.8e+36)
     (fma 0.3333333333333333 (/ t (* z y)) x)
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -600000.0) {
		tmp = fma((1.0 / z), (y / -3.0), x);
	} else if (y <= 6.8e+36) {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -600000.0)
		tmp = fma(Float64(1.0 / z), Float64(y / -3.0), x);
	elseif (y <= 6.8e+36)
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -600000.0], N[(N[(1.0 / z), $MachinePrecision] * N[(y / -3.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 6.8e+36], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6e5

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. metadata-eval 87.7

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

   7. Applied egg-rr 87.7%

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

   if -6e5 < y < 6.7999999999999996e36

   1. Initial program 93.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. /-lowering-/.f64 94.5

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

   5. Simplified 94.5%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-rgt-identity N/A

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

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + 0}{y} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y}} \]

      6. +-rgt-identity N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. associate-/l/ N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. /-lowering-/.f64 N/A

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

      18. *-lowering-*.f64 89.0

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

   8. Simplified 89.0%

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

   if 6.7999999999999996e36 < 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. Step-by-step derivation

      1. associate-+l- N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. sub-div N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -600000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, \frac{y}{-3}, x\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4400:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4400.0)
   (fma (/ y z) -0.3333333333333333 x)
   (if (<= y 5.5e+37)
     (fma 0.3333333333333333 (/ t (* z y)) x)
     (- x (/ y (* z 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4400.0) {
		tmp = fma((y / z), -0.3333333333333333, x);
	} else if (y <= 5.5e+37) {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	} else {
		tmp = x - (y / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4400.0)
		tmp = fma(Float64(y / z), -0.3333333333333333, x);
	elseif (y <= 5.5e+37)
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	else
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4400.0], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 5.5e+37], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4400

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 87.7

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

   7. Applied egg-rr 87.7%

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

   if -4400 < y < 5.50000000000000016e37

   1. Initial program 93.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. /-lowering-/.f64 94.5

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

   5. Simplified 94.5%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-rgt-identity N/A

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

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + 0}{y} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y}} \]

      6. +-rgt-identity N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. associate-/l/ N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. /-lowering-/.f64 N/A

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

      18. *-lowering-*.f64 89.0

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

   8. Simplified 89.0%

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

   if 5.50000000000000016e37 < 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. Step-by-step derivation

      1. associate-+l- N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. sub-div N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4400:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.8e-71)
   (fma (/ y z) -0.3333333333333333 x)
   (if (<= y 3.8e-145)
     (/ t (* (* z 3.0) y))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e-71) {
		tmp = fma((y / z), -0.3333333333333333, x);
	} else if (y <= 3.8e-145) {
		tmp = t / ((z * 3.0) * y);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.8e-71)
		tmp = fma(Float64(y / z), -0.3333333333333333, x);
	elseif (y <= 3.8e-145)
		tmp = Float64(t / Float64(Float64(z * 3.0) * y));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.8e-71], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 3.8e-145], N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e-71

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if -2.8e-71 < y < 3.8000000000000002e-145

   1. Initial program 90.9%

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

   3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 67.7

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

   5. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot t}{y \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. div-inv N/A

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

      7. associate-/l/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 67.8

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

   7. Applied egg-rr 67.8%

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

   if 3.8000000000000002e-145 < y

   1. Initial program 97.9%

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

   3. Taylor expanded in 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 84.4%

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

Alternative 6: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-144}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.65e-70)
   (fma (/ y z) -0.3333333333333333 x)
   (if (<= y 2.5e-144)
     (* t (/ 0.3333333333333333 (* z y)))
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.65e-70) {
		tmp = fma((y / z), -0.3333333333333333, x);
	} else if (y <= 2.5e-144) {
		tmp = t * (0.3333333333333333 / (z * y));
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.65e-70)
		tmp = fma(Float64(y / z), -0.3333333333333333, x);
	elseif (y <= 2.5e-144)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.65e-70], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 2.5e-144], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.65000000000000008e-70

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if -1.65000000000000008e-70 < y < 2.4999999999999999e-144

   1. Initial program 90.9%

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

   3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 67.7

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

   5. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot t}{y \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 67.8

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

   7. Applied egg-rr 67.8%

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

   if 2.4999999999999999e-144 < y

   1. Initial program 97.9%

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

   3. Taylor expanded in 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Simplified 84.4%

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

Alternative 7: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

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

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 N/A

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

   17. --lowering--.f64 N/A

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

   18. /-lowering-/.f64 96.9

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

5. Simplified 96.9%

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

Alternative 8: 47.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e+74) x (if (<= x 0.98) (* y (/ -0.3333333333333333 z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+74) {
		tmp = x;
	} else if (x <= 0.98) {
		tmp = y * (-0.3333333333333333 / z);
	} else {
		tmp = x;
	}
	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 (x <= (-1.1d+74)) then
        tmp = x
    else if (x <= 0.98d0) then
        tmp = y * ((-0.3333333333333333d0) / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+74) {
		tmp = x;
	} else if (x <= 0.98) {
		tmp = y * (-0.3333333333333333 / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e+74:
		tmp = x
	elif x <= 0.98:
		tmp = y * (-0.3333333333333333 / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e+74)
		tmp = x;
	elseif (x <= 0.98)
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e+74)
		tmp = x;
	elseif (x <= 0.98)
		tmp = y * (-0.3333333333333333 / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e+74], x, If[LessEqual[x, 0.98], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e74 or 0.97999999999999998 < x

   1. Initial program 95.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.3%

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

   if -1.1000000000000001e74 < x < 0.97999999999999998

   1. Initial program 96.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   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. *-lowering-*.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. /-lowering-/.f64 48.0

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

   5. Simplified 48.0%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in 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-rgt-in N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

5. Simplified 65.2%

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

Alternative 10: 29.7% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in x around inf

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

5. Simplified 33.5%

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

Developer Target 1: 95.9% 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 2024205 
(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))))