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

Percentage Accurate: 95.5% → 96.7%

Time: 10.8s

Alternatives: 15

Speedup: 1.4×

• 28.3% 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: 95.5% 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: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-197}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 5e-197)
   (+ x (/ (- (/ t y) y) (* z 3.0)))
   (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-197) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	}
	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 ((z * 3.0d0) <= 5d-197) then
        tmp = x + (((t / y) - y) / (z * 3.0d0))
    else
        tmp = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 5e-197) {
		tmp = x + (((t / y) - y) / (z * 3.0));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= 5e-197:
		tmp = x + (((t / y) - y) / (z * 3.0))
	else:
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 5e-197)
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= 5e-197)
		tmp = x + (((t / y) - y) / (z * 3.0));
	else
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-197], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < 5.0000000000000002e-197

   1. Initial program 93.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   if 5.0000000000000002e-197 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 5 \cdot 10^{-197}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ t (* z 3.0)) (/ 1.0 y) (fma (/ y z) -0.3333333333333333 x)))

C

double code(double x, double y, double z, double t) {
	return fma((t / (z * 3.0)), (1.0 / y), fma((y / z), -0.3333333333333333, x));
}

Julia

function code(x, y, z, t)
	return fma(Float64(t / Float64(z * 3.0)), Float64(1.0 / y), fma(Float64(y / z), -0.3333333333333333, x))
end

Wolfram

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

Derivation

1. Initial program 95.9%

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

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

   3. div-inv N/A

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-/r* N/A

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

   11. div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 98.0

       \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{y}}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -7e+16)
     t_1
     (if (<= y 3.5e+40) (fma 0.3333333333333333 (/ (/ t y) z) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -7e+16) {
		tmp = t_1;
	} else if (y <= 3.5e+40) {
		tmp = fma(0.3333333333333333, ((t / y) / z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -7e+16)
		tmp = t_1;
	elseif (y <= 3.5e+40)
		tmp = fma(0.3333333333333333, Float64(Float64(t / y) / z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -7e+16], t$95$1, If[LessEqual[y, 3.5e+40], N[(0.3333333333333333 * N[(N[(t / y), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e16 or 3.4999999999999999e40 < y

   1. Initial program 98.6%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.2

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 94.8

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

   7. Simplified 94.8%

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

   if -7e16 < y < 3.4999999999999999e40

   1. Initial program 93.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. accelerator-lowering-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. /-lowering-/.f64 N/A

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

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

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

      9. /-lowering-/.f64 94.8

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

   5. Simplified 94.8%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 92.6

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

   8. Simplified 92.6%

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

Alternative 4: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -5e+54) x (if (<= (* z 3.0) 5e-52) (/ y (* z -3.0)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -5e+54) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		tmp = y / (z * -3.0);
	} 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 ((z * 3.0d0) <= (-5d+54)) then
        tmp = x
    else if ((z * 3.0d0) <= 5d-52) then
        tmp = y / (z * (-3.0d0))
    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 ((z * 3.0) <= -5e+54) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		tmp = y / (z * -3.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -5e+54:
		tmp = x
	elif (z * 3.0) <= 5e-52:
		tmp = y / (z * -3.0)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -5e+54)
		tmp = x;
	elseif (Float64(z * 3.0) <= 5e-52)
		tmp = Float64(y / Float64(z * -3.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= -5e+54)
		tmp = x;
	elseif ((z * 3.0) <= 5e-52)
		tmp = y / (z * -3.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -5e+54], x, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-52], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -5.00000000000000005e54 or 5e-52 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 64.7%

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

   if -5.00000000000000005e54 < (*.f64 z #s(literal 3 binary64)) < 5e-52

   1. Initial program 91.8%

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.0

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in y around inf

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

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

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

      2. /-lowering-/.f64 53.3

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

   7. Simplified 53.3%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. neg-mul-1 N/A

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-frac N/A

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

      10. remove-double-neg N/A

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

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

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

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

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

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

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

      14. metadata-eval 53.4

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

   9. Applied egg-rr 53.4%

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

Alternative 5: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -8 \cdot 10^{+46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -8e+46)
   x
   (if (<= (* z 3.0) 5e-52) (* y (/ -0.3333333333333333 z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -8e+46) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		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 ((z * 3.0d0) <= (-8d+46)) then
        tmp = x
    else if ((z * 3.0d0) <= 5d-52) 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 ((z * 3.0) <= -8e+46) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		tmp = y * (-0.3333333333333333 / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -8e+46:
		tmp = x
	elif (z * 3.0) <= 5e-52:
		tmp = y * (-0.3333333333333333 / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -8e+46)
		tmp = x;
	elseif (Float64(z * 3.0) <= 5e-52)
		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 ((z * 3.0) <= -8e+46)
		tmp = x;
	elseif ((z * 3.0) <= 5e-52)
		tmp = y * (-0.3333333333333333 / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -8e+46], x, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-52], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -7.9999999999999999e46 or 5e-52 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 64.5%

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

   if -7.9999999999999999e46 < (*.f64 z #s(literal 3 binary64)) < 5e-52

   1. Initial program 91.7%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.0

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in y around inf

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

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

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

      2. /-lowering-/.f64 53.3

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

   7. Simplified 53.3%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. /-rgt-identity N/A

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

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

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

      10. /-lowering-/.f64 53.3

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

   9. Applied egg-rr 53.3%

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

Alternative 6: 47.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -5e+54)
   x
   (if (<= (* z 3.0) 5e-52) (* (/ y z) -0.3333333333333333) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -5e+54) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		tmp = (y / z) * -0.3333333333333333;
	} 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 ((z * 3.0d0) <= (-5d+54)) then
        tmp = x
    else if ((z * 3.0d0) <= 5d-52) then
        tmp = (y / z) * (-0.3333333333333333d0)
    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 ((z * 3.0) <= -5e+54) {
		tmp = x;
	} else if ((z * 3.0) <= 5e-52) {
		tmp = (y / z) * -0.3333333333333333;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -5e+54:
		tmp = x
	elif (z * 3.0) <= 5e-52:
		tmp = (y / z) * -0.3333333333333333
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -5e+54)
		tmp = x;
	elseif (Float64(z * 3.0) <= 5e-52)
		tmp = Float64(Float64(y / z) * -0.3333333333333333);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= -5e+54)
		tmp = x;
	elseif ((z * 3.0) <= 5e-52)
		tmp = (y / z) * -0.3333333333333333;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -5e+54], x, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-52], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -5.00000000000000005e54 or 5e-52 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 64.7%

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

   if -5.00000000000000005e54 < (*.f64 z #s(literal 3 binary64)) < 5e-52

   1. Initial program 91.8%

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.0

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in y around inf

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

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

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

      2. /-lowering-/.f64 53.3

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

   7. Simplified 53.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -6.5e+18)
     t_1
     (if (<= y 3.2e+40) (+ x (/ t (* (* z 3.0) y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -6.5e+18) {
		tmp = t_1;
	} else if (y <= 3.2e+40) {
		tmp = x + (t / ((z * 3.0) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -6.5e+18)
		tmp = t_1;
	elseif (y <= 3.2e+40)
		tmp = Float64(x + Float64(t / Float64(Float64(z * 3.0) * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -6.5e+18], t$95$1, If[LessEqual[y, 3.2e+40], N[(x + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5e18 or 3.19999999999999981e40 < y

   1. Initial program 98.6%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.2

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 94.8

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

   7. Simplified 94.8%

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

   if -6.5e18 < y < 3.19999999999999981e40

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 91.8%

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

Alternative 8: 89.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -5e+18)
     t_1
     (if (<= y 3.9e+40) (fma 0.3333333333333333 (/ t (* z y)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -5e+18) {
		tmp = t_1;
	} else if (y <= 3.9e+40) {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -5e+18)
		tmp = t_1;
	elseif (y <= 3.9e+40)
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5e+18], t$95$1, If[LessEqual[y, 3.9e+40], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e18 or 3.9000000000000001e40 < y

   1. Initial program 98.6%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.2

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 94.8

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

   7. Simplified 94.8%

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

   if -5e18 < y < 3.9000000000000001e40

   1. Initial program 93.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. accelerator-lowering-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. /-lowering-/.f64 N/A

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

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

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

      9. /-lowering-/.f64 94.8

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

   5. Simplified 94.8%

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

   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 91.7

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

   8. Simplified 91.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7e-157)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 3.8e-113)
     (/ t (* (* z 3.0) y))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-157) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 3.8e-113) {
		tmp = t / ((z * 3.0) * y);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7e-157)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 3.8e-113)
		tmp = Float64(t / Float64(Float64(z * 3.0) * y));
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7e-157], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.8e-113], N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000004e-157

   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 81.5%

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

   if -7.0000000000000004e-157 < y < 3.79999999999999983e-113

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 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 60.2

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

   5. Simplified 60.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 60.2

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

   7. Applied egg-rr 60.2%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. *-lft-identity N/A

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 60.4

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

   9. Applied egg-rr 60.4%

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

   if 3.79999999999999983e-113 < y

   1. Initial program 97.4%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 83.9

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

   7. Simplified 83.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5e-157)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 1.8e-113)
     (/ t (* 3.0 (* z y)))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e-157) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 1.8e-113) {
		tmp = t / (3.0 * (z * y));
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5e-157)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 1.8e-113)
		tmp = Float64(t / Float64(3.0 * 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, -5e-157], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.8e-113], N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000002e-157

   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 81.5%

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

   if -5.0000000000000002e-157 < y < 1.79999999999999987e-113

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 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 60.2

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

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot t}{y \cdot z}} \]
   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. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 60.3

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

   7. Applied egg-rr 60.3%

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

   if 1.79999999999999987e-113 < y

   1. Initial program 97.4%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 83.9

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

   7. Simplified 83.9%

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

Alternative 11: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-113}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.8e-157)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 5e-113)
     (* 0.3333333333333333 (/ t (* z y)))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.8e-157) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 5e-113) {
		tmp = 0.3333333333333333 * (t / (z * y));
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.8e-157)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 5e-113)
		tmp = Float64(0.3333333333333333 * Float64(t / 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, -5.8e-157], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 5e-113], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999977e-157

   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 81.5%

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

   if -5.79999999999999977e-157 < y < 4.9999999999999997e-113

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 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 60.2

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

   5. Simplified 60.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 60.2

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

   7. Applied egg-rr 60.2%

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

   if 4.9999999999999997e-113 < y

   1. Initial program 97.4%

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

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

      3. div-inv N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 98.7

          \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. /-lowering-/.f64 83.9

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

   7. Simplified 83.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-113}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 + (((t / y) - y) / (z * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 97.3

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

4. Applied egg-rr 97.3%

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

5. Final simplification 97.3%

   \[\leadsto x + \frac{\frac{t}{y} - y}{z \cdot 3} \]
6. Add Preprocessing

Alternative 13: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. accelerator-lowering-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. /-lowering-/.f64 N/A

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

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

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

   9. /-lowering-/.f64 97.2

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

5. Simplified 97.2%

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

Alternative 14: 64.3% 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 95.9%

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

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

   3. div-inv N/A

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-/r* N/A

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

   11. div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 98.0

       \[\leadsto \mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, \color{blue}{-0.3333333333333333}, x\right)\right) \]

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot 3}, \frac{1}{y}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

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

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

   3. /-lowering-/.f64 71.3

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

7. Simplified 71.3%

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

Alternative 15: 30.6% 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 95.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 38.4%

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

Developer Target 1: 95.8% 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 2024195 
(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))))