Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 9

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(0.125, x, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma z (/ y -2.0) (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	return fma(z, (y / -2.0), fma(0.125, x, t));
}

Julia

function code(x, y, z, t)
	return fma(z, Float64(y / -2.0), fma(0.125, x, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(z * N[(y / -2.0), $MachinePrecision] + N[(0.125 * x + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. remove-double-neg 99.7%

      \[\leadsto \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + \color{blue}{\left(-\left(-t\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) - \left(-t\right)} \]

   3. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} - \left(-t\right) \]

   4. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} - \left(-t\right) \]

   5. associate--l+ 99.7%

      \[\leadsto \color{blue}{\left(-\frac{y \cdot z}{2}\right) + \left(\frac{1}{8} \cdot x - \left(-t\right)\right)} \]

   6. *-commutative 99.7%

      \[\leadsto \left(-\frac{\color{blue}{z \cdot y}}{2}\right) + \left(\frac{1}{8} \cdot x - \left(-t\right)\right) \]

   7. associate-*r/ 100.0%

      \[\leadsto \left(-\color{blue}{z \cdot \frac{y}{2}}\right) + \left(\frac{1}{8} \cdot x - \left(-t\right)\right) \]

   8. distribute-rgt-neg-in 100.0%

      \[\leadsto \color{blue}{z \cdot \left(-\frac{y}{2}\right)} + \left(\frac{1}{8} \cdot x - \left(-t\right)\right) \]

   9. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, -\frac{y}{2}, \frac{1}{8} \cdot x - \left(-t\right)\right)} \]

   10. neg-mul-1 100.0%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{-1 \cdot \frac{y}{2}}, \frac{1}{8} \cdot x - \left(-t\right)\right) \]

   11. *-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y}{2} \cdot -1}, \frac{1}{8} \cdot x - \left(-t\right)\right) \]

   12. associate-*l/ 100.0%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y \cdot -1}{2}}, \frac{1}{8} \cdot x - \left(-t\right)\right) \]

   13. associate-/l* 100.0%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y}{\frac{2}{-1}}}, \frac{1}{8} \cdot x - \left(-t\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(z, \frac{y}{\color{blue}{-2}}, \frac{1}{8} \cdot x - \left(-t\right)\right) \]

   15. fma-neg 100.0%

       \[\leadsto \mathsf{fma}\left(z, \frac{y}{-2}, \color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, -\left(-t\right)\right)}\right) \]

   16. remove-double-neg 100.0%

       \[\leadsto \mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(\frac{1}{8}, x, \color{blue}{t}\right)\right) \]

   17. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(\color{blue}{0.125}, x, t\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(0.125, x, t\right)\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(0.125, x, t\right)\right) \]

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ t + \mathsf{fma}\left(y \cdot -0.5, z, 0.125 \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (fma (* y -0.5) z (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	return t + fma((y * -0.5), z, (0.125 * x));
}

Julia

function code(x, y, z, t)
	return Float64(t + fma(Float64(y * -0.5), z, Float64(0.125 * x)))
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(y * -0.5), $MachinePrecision] * z + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

   2. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

   3. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

   4. associate-+l- 99.7%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

   5. sub-neg 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

   6. +-commutative 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

   7. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

   8. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   9. distribute-rgt-neg-in 99.7%

      \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

   10. distribute-rgt-neg-in 99.7%

       \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   11. metadata-eval 99.7%

       \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

   12. remove-double-neg 99.7%

       \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

   13. associate-*l/ 100.0%

       \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]
4. Step-by-step derivation

   1. associate-/r/ 99.9%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   2. sub-neg 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x + \left(-\frac{y}{\frac{2}{z}}\right)\right)} + t \]

   3. +-commutative 99.9%

      \[\leadsto \color{blue}{\left(\left(-\frac{y}{\frac{2}{z}}\right) + 0.125 \cdot x\right)} + t \]

   4. associate-/r/ 100.0%

      \[\leadsto \left(\left(-\color{blue}{\frac{y}{2} \cdot z}\right) + 0.125 \cdot x\right) + t \]

   5. distribute-lft-neg-in 100.0%

      \[\leadsto \left(\color{blue}{\left(-\frac{y}{2}\right) \cdot z} + 0.125 \cdot x\right) + t \]

   6. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{y}{2}, z, 0.125 \cdot x\right)} + t \]

   7. div-inv 100.0%

      \[\leadsto \mathsf{fma}\left(-\color{blue}{y \cdot \frac{1}{2}}, z, 0.125 \cdot x\right) + t \]

   8. distribute-rgt-neg-in 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(-\frac{1}{2}\right)}, z, 0.125 \cdot x\right) + t \]

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y \cdot \left(-\color{blue}{0.5}\right), z, 0.125 \cdot x\right) + t \]

   10. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{-0.5}, z, 0.125 \cdot x\right) + t \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot -0.5, z, 0.125 \cdot x\right)} + t \]

6. Final simplification 100.0%

   \[\leadsto t + \mathsf{fma}\left(y \cdot -0.5, z, 0.125 \cdot x\right) \]

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+42} \lor \neg \left(z \cdot y \leq 5 \cdot 10^{-44}\right):\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -1e+42) (not (<= (* z y) 5e-44)))
   (+ t (* z (/ y -2.0)))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+42) || !((z * y) <= 5e-44)) {
		tmp = t + (z * (y / -2.0));
	} else {
		tmp = t + (0.125 * 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 * y) <= (-1d+42)) .or. (.not. ((z * y) <= 5d-44))) then
        tmp = t + (z * (y / (-2.0d0)))
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+42) || !((z * y) <= 5e-44)) {
		tmp = t + (z * (y / -2.0));
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -1e+42) or not ((z * y) <= 5e-44):
		tmp = t + (z * (y / -2.0))
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -1e+42) || !(Float64(z * y) <= 5e-44))
		tmp = Float64(t + Float64(z * Float64(y / -2.0)));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -1e+42) || ~(((z * y) <= 5e-44)))
		tmp = t + (z * (y / -2.0));
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -1e+42], N[Not[LessEqual[N[(z * y), $MachinePrecision], 5e-44]], $MachinePrecision]], N[(t + N[(z * N[(y / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000004e42 or 5.00000000000000039e-44 < (*.f64 y z)

   1. Initial program 99.3%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 99.3%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 99.3%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 99.3%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 99.3%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 99.3%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 99.3%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 99.3%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 99.3%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 99.3%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 99.3%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 99.3%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 99.9%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around 0 88.4%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} + t \]
   5. Step-by-step derivation

      1. associate-*r* 89.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot y\right) \cdot z} + t \]

      2. *-commutative 89.0%

         \[\leadsto \color{blue}{\left(y \cdot -0.5\right)} \cdot z + t \]

      3. associate-*r* 89.0%

         \[\leadsto \color{blue}{y \cdot \left(-0.5 \cdot z\right)} + t \]

      4. metadata-eval 89.0%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-2}} \cdot z\right) + t \]

      5. associate-/r/ 89.0%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{-2}{z}}} + t \]

      6. associate-*r/ 88.8%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\frac{-2}{z}}} + t \]

      7. *-rgt-identity 88.8%

         \[\leadsto \frac{\color{blue}{y}}{\frac{-2}{z}} + t \]

      8. associate-/r/ 89.0%

         \[\leadsto \color{blue}{\frac{y}{-2} \cdot z} + t \]

      9. *-commutative 89.0%

         \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   6. Simplified 89.0%

      \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   if -1.00000000000000004e42 < (*.f64 y z) < 5.00000000000000039e-44

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 100.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 100.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around inf 93.5%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+42} \lor \neg \left(z \cdot y \leq 5 \cdot 10^{-44}\right):\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 4: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+89}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z y) -1e+89)
   (- (* 0.125 x) (* (* z y) 0.5))
   (if (<= (* z y) 5e-44) (+ t (* 0.125 x)) (+ t (* z (/ y -2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * y) <= -1e+89) {
		tmp = (0.125 * x) - ((z * y) * 0.5);
	} else if ((z * y) <= 5e-44) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t + (z * (y / -2.0));
	}
	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 * y) <= (-1d+89)) then
        tmp = (0.125d0 * x) - ((z * y) * 0.5d0)
    else if ((z * y) <= 5d-44) then
        tmp = t + (0.125d0 * x)
    else
        tmp = t + (z * (y / (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * y) <= -1e+89) {
		tmp = (0.125 * x) - ((z * y) * 0.5);
	} else if ((z * y) <= 5e-44) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t + (z * (y / -2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * y) <= -1e+89:
		tmp = (0.125 * x) - ((z * y) * 0.5)
	elif (z * y) <= 5e-44:
		tmp = t + (0.125 * x)
	else:
		tmp = t + (z * (y / -2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * y) <= -1e+89)
		tmp = Float64(Float64(0.125 * x) - Float64(Float64(z * y) * 0.5));
	elseif (Float64(z * y) <= 5e-44)
		tmp = Float64(t + Float64(0.125 * x));
	else
		tmp = Float64(t + Float64(z * Float64(y / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * y) <= -1e+89)
		tmp = (0.125 * x) - ((z * y) * 0.5);
	elseif ((z * y) <= 5e-44)
		tmp = t + (0.125 * x);
	else
		tmp = t + (z * (y / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * y), $MachinePrecision], -1e+89], N[(N[(0.125 * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 5e-44], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(t + N[(z * N[(y / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -9.99999999999999995e88

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 99.9%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 99.9%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 99.9%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 99.9%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 99.9%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 99.9%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 99.9%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 99.9%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 99.9%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in t around 0 95.1%

      \[\leadsto \color{blue}{0.125 \cdot x - 0.5 \cdot \left(y \cdot z\right)} \]

   if -9.99999999999999995e88 < (*.f64 y z) < 5.00000000000000039e-44

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 100.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 100.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around inf 92.3%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]

   if 5.00000000000000039e-44 < (*.f64 y z)

   1. Initial program 98.8%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 98.8%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 98.8%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 98.8%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 98.8%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 98.8%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 98.8%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 98.8%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 98.8%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 98.8%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 98.8%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 98.8%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 98.8%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 99.9%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around 0 86.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} + t \]
   5. Step-by-step derivation

      1. associate-*r* 87.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot y\right) \cdot z} + t \]

      2. *-commutative 87.5%

         \[\leadsto \color{blue}{\left(y \cdot -0.5\right)} \cdot z + t \]

      3. associate-*r* 87.5%

         \[\leadsto \color{blue}{y \cdot \left(-0.5 \cdot z\right)} + t \]

      4. metadata-eval 87.5%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-2}} \cdot z\right) + t \]

      5. associate-/r/ 87.4%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{-2}{z}}} + t \]

      6. associate-*r/ 87.3%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\frac{-2}{z}}} + t \]

      7. *-rgt-identity 87.3%

         \[\leadsto \frac{\color{blue}{y}}{\frac{-2}{z}} + t \]

      8. associate-/r/ 87.5%

         \[\leadsto \color{blue}{\frac{y}{-2} \cdot z} + t \]

      9. *-commutative 87.5%

         \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   6. Simplified 87.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+89}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \end{array} \]

Alternative 5: 56.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1.6 \cdot 10^{+40} \lor \neg \left(z \cdot y \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -1.6e+40) (not (<= (* z y) 2.55e-23)))
   (* -0.5 (* z y))
   (* 0.125 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1.6e+40) || !((z * y) <= 2.55e-23)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = 0.125 * 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 * y) <= (-1.6d+40)) .or. (.not. ((z * y) <= 2.55d-23))) then
        tmp = (-0.5d0) * (z * y)
    else
        tmp = 0.125d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1.6e+40) || !((z * y) <= 2.55e-23)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = 0.125 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -1.6e+40) or not ((z * y) <= 2.55e-23):
		tmp = -0.5 * (z * y)
	else:
		tmp = 0.125 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -1.6e+40) || !(Float64(z * y) <= 2.55e-23))
		tmp = Float64(-0.5 * Float64(z * y));
	else
		tmp = Float64(0.125 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -1.6e+40) || ~(((z * y) <= 2.55e-23)))
		tmp = -0.5 * (z * y);
	else
		tmp = 0.125 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -1.6e+40], N[Not[LessEqual[N[(z * y), $MachinePrecision], 2.55e-23]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(0.125 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.5999999999999999e40 or 2.55000000000000005e-23 < (*.f64 y z)

   1. Initial program 99.3%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 99.3%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 99.3%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 99.3%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 99.3%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 99.3%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 99.3%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 99.3%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 99.3%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 99.3%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 99.3%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 99.3%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 99.9%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around 0 88.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} + t \]
   5. Step-by-step derivation

      1. associate-*r* 89.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot y\right) \cdot z} + t \]

      2. *-commutative 89.3%

         \[\leadsto \color{blue}{\left(y \cdot -0.5\right)} \cdot z + t \]

      3. associate-*r* 89.3%

         \[\leadsto \color{blue}{y \cdot \left(-0.5 \cdot z\right)} + t \]

      4. metadata-eval 89.3%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-2}} \cdot z\right) + t \]

      5. associate-/r/ 89.3%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{-2}{z}}} + t \]

      6. associate-*r/ 89.1%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\frac{-2}{z}}} + t \]

      7. *-rgt-identity 89.1%

         \[\leadsto \frac{\color{blue}{y}}{\frac{-2}{z}} + t \]

      8. associate-/r/ 89.3%

         \[\leadsto \color{blue}{\frac{y}{-2} \cdot z} + t \]

      9. *-commutative 89.3%

         \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   6. Simplified 89.3%

      \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   7. Taylor expanded in z around inf 73.4%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} \]

   if -1.5999999999999999e40 < (*.f64 y z) < 2.55000000000000005e-23

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 100.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 100.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.125 \cdot x + \left(-\frac{y}{\frac{2}{z}}\right)\right)} + t \]

      3. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y}{\frac{2}{z}}\right) + 0.125 \cdot x\right)} + t \]

      4. associate-/r/ 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{y}{2} \cdot z}\right) + 0.125 \cdot x\right) + t \]

      5. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-\frac{y}{2}\right) \cdot z} + 0.125 \cdot x\right) + t \]

      6. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{y}{2}, z, 0.125 \cdot x\right)} + t \]

      7. div-inv 100.0%

         \[\leadsto \mathsf{fma}\left(-\color{blue}{y \cdot \frac{1}{2}}, z, 0.125 \cdot x\right) + t \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(-\frac{1}{2}\right)}, z, 0.125 \cdot x\right) + t \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(-\color{blue}{0.5}\right), z, 0.125 \cdot x\right) + t \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{-0.5}, z, 0.125 \cdot x\right) + t \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot -0.5, z, 0.125 \cdot x\right)} + t \]

   6. Taylor expanded in x around inf 55.7%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1.6 \cdot 10^{+40} \lor \neg \left(z \cdot y \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x\\ \end{array} \]

Alternative 6: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2.9 \cdot 10^{+87} \lor \neg \left(z \cdot y \leq 1.1 \cdot 10^{+157}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -2.9e+87) (not (<= (* z y) 1.1e+157)))
   (* -0.5 (* z y))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -2.9e+87) || !((z * y) <= 1.1e+157)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = t + (0.125 * 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 * y) <= (-2.9d+87)) .or. (.not. ((z * y) <= 1.1d+157))) then
        tmp = (-0.5d0) * (z * y)
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -2.9e+87) || !((z * y) <= 1.1e+157)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -2.9e+87) or not ((z * y) <= 1.1e+157):
		tmp = -0.5 * (z * y)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -2.9e+87) || !(Float64(z * y) <= 1.1e+157))
		tmp = Float64(-0.5 * Float64(z * y));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -2.9e+87) || ~(((z * y) <= 1.1e+157)))
		tmp = -0.5 * (z * y);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -2.9e+87], N[Not[LessEqual[N[(z * y), $MachinePrecision], 1.1e+157]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2.8999999999999998e87 or 1.1000000000000001e157 < (*.f64 y z)

   1. Initial program 99.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 99.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 99.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 99.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 99.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 99.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 99.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 99.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 99.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 99.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 99.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 99.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 99.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 99.9%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around 0 92.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} + t \]
   5. Step-by-step derivation

      1. associate-*r* 93.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot y\right) \cdot z} + t \]

      2. *-commutative 93.3%

         \[\leadsto \color{blue}{\left(y \cdot -0.5\right)} \cdot z + t \]

      3. associate-*r* 93.3%

         \[\leadsto \color{blue}{y \cdot \left(-0.5 \cdot z\right)} + t \]

      4. metadata-eval 93.3%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-2}} \cdot z\right) + t \]

      5. associate-/r/ 93.2%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{-2}{z}}} + t \]

      6. associate-*r/ 93.1%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\frac{-2}{z}}} + t \]

      7. *-rgt-identity 93.1%

         \[\leadsto \frac{\color{blue}{y}}{\frac{-2}{z}} + t \]

      8. associate-/r/ 93.3%

         \[\leadsto \color{blue}{\frac{y}{-2} \cdot z} + t \]

      9. *-commutative 93.3%

         \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   6. Simplified 93.3%

      \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   7. Taylor expanded in z around inf 88.2%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} \]

   if -2.8999999999999998e87 < (*.f64 y z) < 1.1000000000000001e157

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 100.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 100.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in x around inf 85.9%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2.9 \cdot 10^{+87} \lor \neg \left(z \cdot y \leq 1.1 \cdot 10^{+157}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ t + \left(0.125 \cdot x - z \cdot \frac{y}{2}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (- (* 0.125 x) (* z (/ y 2.0)))))

C

double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (z * (y / 2.0)));
}

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 = t + ((0.125d0 * x) - (z * (y / 2.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (z * (y / 2.0)));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - (z * (y / 2.0)))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(0.125 * x) - Float64(z * Float64(y / 2.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - (z * (y / 2.0)));
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(0.125 * x), $MachinePrecision] - N[(z * N[(y / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

   2. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

   3. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

   4. associate-+l- 99.7%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

   5. sub-neg 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

   6. +-commutative 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

   7. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

   8. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   9. distribute-rgt-neg-in 99.7%

      \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

   10. distribute-rgt-neg-in 99.7%

       \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   11. metadata-eval 99.7%

       \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

   12. remove-double-neg 99.7%

       \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

   13. associate-*l/ 100.0%

       \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

4. Final simplification 100.0%

   \[\leadsto t + \left(0.125 \cdot x - z \cdot \frac{y}{2}\right) \]

Alternative 8: 49.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+75}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+22}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.2e+75) t (if (<= t 3.3e+22) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.2e+75) {
		tmp = t;
	} else if (t <= 3.3e+22) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	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 (t <= (-7.2d+75)) then
        tmp = t
    else if (t <= 3.3d+22) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.2e+75) {
		tmp = t;
	} else if (t <= 3.3e+22) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.2e+75:
		tmp = t
	elif t <= 3.3e+22:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.2e+75)
		tmp = t;
	elseif (t <= 3.3e+22)
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.2e+75)
		tmp = t;
	elseif (t <= 3.3e+22)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.2e+75], t, If[LessEqual[t, 3.3e+22], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2e75 or 3.2999999999999998e22 < t

   1. Initial program 99.2%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 99.2%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 99.2%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 99.2%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 99.2%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 99.2%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 99.2%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 99.2%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 99.2%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 99.2%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 99.2%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 99.2%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

   4. Taylor expanded in t around inf 59.7%

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

   if -7.2e75 < t < 3.2999999999999998e22

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      9. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

      11. metadata-eval 100.0%

          \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

      12. remove-double-neg 100.0%

          \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

      13. associate-*l/ 100.0%

          \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]
   4. Step-by-step derivation

      1. associate-/r/ 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

      2. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(0.125 \cdot x + \left(-\frac{y}{\frac{2}{z}}\right)\right)} + t \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\left(-\frac{y}{\frac{2}{z}}\right) + 0.125 \cdot x\right)} + t \]

      4. associate-/r/ 100.0%

         \[\leadsto \left(\left(-\color{blue}{\frac{y}{2} \cdot z}\right) + 0.125 \cdot x\right) + t \]

      5. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-\frac{y}{2}\right) \cdot z} + 0.125 \cdot x\right) + t \]

      6. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{y}{2}, z, 0.125 \cdot x\right)} + t \]

      7. div-inv 100.0%

         \[\leadsto \mathsf{fma}\left(-\color{blue}{y \cdot \frac{1}{2}}, z, 0.125 \cdot x\right) + t \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(-\frac{1}{2}\right)}, z, 0.125 \cdot x\right) + t \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(-\color{blue}{0.5}\right), z, 0.125 \cdot x\right) + t \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{-0.5}, z, 0.125 \cdot x\right) + t \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot -0.5, z, 0.125 \cdot x\right)} + t \]

   6. Taylor expanded in x around inf 46.1%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+75}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+22}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9: 32.7% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(-\frac{y \cdot z}{2}\right)\right)} + t \]

   2. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(-\frac{y \cdot z}{2}\right) + \frac{1}{8} \cdot x\right)} + t \]

   3. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

   4. associate-+l- 99.7%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

   5. sub-neg 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

   6. +-commutative 99.7%

      \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

   7. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

   8. neg-sub0 99.7%

      \[\leadsto \left(\color{blue}{\left(-\left(-\frac{1}{8} \cdot x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   9. distribute-rgt-neg-in 99.7%

      \[\leadsto \left(\left(-\color{blue}{\frac{1}{8} \cdot \left(-x\right)}\right) - \frac{y \cdot z}{2}\right) + t \]

   10. distribute-rgt-neg-in 99.7%

       \[\leadsto \left(\color{blue}{\frac{1}{8} \cdot \left(-\left(-x\right)\right)} - \frac{y \cdot z}{2}\right) + t \]

   11. metadata-eval 99.7%

       \[\leadsto \left(\color{blue}{0.125} \cdot \left(-\left(-x\right)\right) - \frac{y \cdot z}{2}\right) + t \]

   12. remove-double-neg 99.7%

       \[\leadsto \left(0.125 \cdot \color{blue}{x} - \frac{y \cdot z}{2}\right) + t \]

   13. associate-*l/ 100.0%

       \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{2} \cdot z}\right) + t \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{2} \cdot z\right) + t} \]

4. Taylor expanded in t around inf 28.1%

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

5. Final simplification 28.1%

   \[\leadsto t \]

Developer target: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (+ (/ x 8.0) t) (* (/ z 2.0) y)))

C

double code(double x, double y, double z, double t) {
	return ((x / 8.0) + t) - ((z / 2.0) * 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 / 8.0d0) + t) - ((z / 2.0d0) * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x / 8.0) + t) - ((z / 2.0) * y);
}

Python

def code(x, y, z, t):
	return ((x / 8.0) + t) - ((z / 2.0) * y)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / 8.0) + t) - Float64(Float64(z / 2.0) * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / 8.0) + t) - ((z / 2.0) * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / 8.0), $MachinePrecision] + t), $MachinePrecision] - N[(N[(z / 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))