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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

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 100.0%

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

   1. remove-double-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. associate--l+ 100.0%

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

   6. *-commutative 100.0%

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

   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 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. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-neg-in 100.0%

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

   4. fma-def 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

   8. 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: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot 0.5\\ t_2 := 0.125 \cdot x - t_1\\ \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;t - t_1\\ \mathbf{elif}\;z \cdot y \leq 10^{+71}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z y) 0.5)) (t_2 (- (* 0.125 x) t_1)))
   (if (<= (* z y) -2e+92)
     t_2
     (if (<= (* z y) -2e-19)
       (- t t_1)
       (if (<= (* z y) 1e+71) (+ t (* 0.125 x)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) * 0.5;
	double t_2 = (0.125 * x) - t_1;
	double tmp;
	if ((z * y) <= -2e+92) {
		tmp = t_2;
	} else if ((z * y) <= -2e-19) {
		tmp = t - t_1;
	} else if ((z * y) <= 1e+71) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * y) * 0.5d0
    t_2 = (0.125d0 * x) - t_1
    if ((z * y) <= (-2d+92)) then
        tmp = t_2
    else if ((z * y) <= (-2d-19)) then
        tmp = t - t_1
    else if ((z * y) <= 1d+71) then
        tmp = t + (0.125d0 * x)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) * 0.5;
	double t_2 = (0.125 * x) - t_1;
	double tmp;
	if ((z * y) <= -2e+92) {
		tmp = t_2;
	} else if ((z * y) <= -2e-19) {
		tmp = t - t_1;
	} else if ((z * y) <= 1e+71) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) * 0.5
	t_2 = (0.125 * x) - t_1
	tmp = 0
	if (z * y) <= -2e+92:
		tmp = t_2
	elif (z * y) <= -2e-19:
		tmp = t - t_1
	elif (z * y) <= 1e+71:
		tmp = t + (0.125 * x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) * 0.5)
	t_2 = Float64(Float64(0.125 * x) - t_1)
	tmp = 0.0
	if (Float64(z * y) <= -2e+92)
		tmp = t_2;
	elseif (Float64(z * y) <= -2e-19)
		tmp = Float64(t - t_1);
	elseif (Float64(z * y) <= 1e+71)
		tmp = Float64(t + Float64(0.125 * x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) * 0.5;
	t_2 = (0.125 * x) - t_1;
	tmp = 0.0;
	if ((z * y) <= -2e+92)
		tmp = t_2;
	elseif ((z * y) <= -2e-19)
		tmp = t - t_1;
	elseif ((z * y) <= 1e+71)
		tmp = t + (0.125 * x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.125 * x), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -2e+92], t$95$2, If[LessEqual[N[(z * y), $MachinePrecision], -2e-19], N[(t - t$95$1), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 1e+71], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -2.0000000000000001e92 or 1e71 < (*.f64 y z)

   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 93.5%

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

   if -2.0000000000000001e92 < (*.f64 y z) < -2e-19

   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 0 93.8%

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

   if -2e-19 < (*.f64 y z) < 1e71

   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 93.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{+92}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;z \cdot y \leq -2 \cdot 10^{-19}:\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;z \cdot y \leq 10^{+71}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \end{array} \]

Alternative 4: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+234} \lor \neg \left(y \leq -9.8 \cdot 10^{+153} \lor \neg \left(y \leq -3.1 \cdot 10^{+121}\right) \land \left(y \leq -4.8 \cdot 10^{+93} \lor \neg \left(y \leq -9.2 \cdot 10^{+42}\right) \land y \leq 5.4 \cdot 10^{-75}\right)\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.5e+234)
         (not
          (or (<= y -9.8e+153)
              (and (not (<= y -3.1e+121))
                   (or (<= y -4.8e+93)
                       (and (not (<= y -9.2e+42)) (<= y 5.4e-75)))))))
   (* y (* z -0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.5e+234) || !((y <= -9.8e+153) || (!(y <= -3.1e+121) && ((y <= -4.8e+93) || (!(y <= -9.2e+42) && (y <= 5.4e-75)))))) {
		tmp = y * (z * -0.5);
	} 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 ((y <= (-5.5d+234)) .or. (.not. (y <= (-9.8d+153)) .or. (.not. (y <= (-3.1d+121))) .and. (y <= (-4.8d+93)) .or. (.not. (y <= (-9.2d+42))) .and. (y <= 5.4d-75))) then
        tmp = y * (z * (-0.5d0))
    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 ((y <= -5.5e+234) || !((y <= -9.8e+153) || (!(y <= -3.1e+121) && ((y <= -4.8e+93) || (!(y <= -9.2e+42) && (y <= 5.4e-75)))))) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.5e+234) or not ((y <= -9.8e+153) or (not (y <= -3.1e+121) and ((y <= -4.8e+93) or (not (y <= -9.2e+42) and (y <= 5.4e-75))))):
		tmp = y * (z * -0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.5e+234) || !((y <= -9.8e+153) || (!(y <= -3.1e+121) && ((y <= -4.8e+93) || (!(y <= -9.2e+42) && (y <= 5.4e-75))))))
		tmp = Float64(y * Float64(z * -0.5));
	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 ((y <= -5.5e+234) || ~(((y <= -9.8e+153) || (~((y <= -3.1e+121)) && ((y <= -4.8e+93) || (~((y <= -9.2e+42)) && (y <= 5.4e-75)))))))
		tmp = y * (z * -0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.5e+234], N[Not[Or[LessEqual[y, -9.8e+153], And[N[Not[LessEqual[y, -3.1e+121]], $MachinePrecision], Or[LessEqual[y, -4.8e+93], And[N[Not[LessEqual[y, -9.2e+42]], $MachinePrecision], LessEqual[y, 5.4e-75]]]]]], $MachinePrecision]], N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e234 or -9.80000000000000003e153 < y < -3.10000000000000008e121 or -4.80000000000000021e93 < y < -9.2e42 or 5.3999999999999996e-75 < y

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. div-inv 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. 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 y around inf 60.1%

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

      1. associate-*r* 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*r* 60.1%

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

   8. Simplified 60.1%

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

   if -5.5e234 < y < -9.80000000000000003e153 or -3.10000000000000008e121 < y < -4.80000000000000021e93 or -9.2e42 < y < 5.3999999999999996e-75

   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 81.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+234} \lor \neg \left(y \leq -9.8 \cdot 10^{+153} \lor \neg \left(y \leq -3.1 \cdot 10^{+121}\right) \land \left(y \leq -4.8 \cdot 10^{+93} \lor \neg \left(y \leq -9.2 \cdot 10^{+42}\right) \land y \leq 5.4 \cdot 10^{-75}\right)\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 5: 48.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z \cdot -0.5\right)\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-34}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-273}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* z -0.5))))
   (if (<= y -2.05e+42)
     t_1
     (if (<= y -1.3e-34)
       (* 0.125 x)
       (if (<= y -4.2e-273)
         t
         (if (<= y 6.8e-274) (* 0.125 x) (if (<= y 1.85e-75) t t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if (y <= -2.05e+42) {
		tmp = t_1;
	} else if (y <= -1.3e-34) {
		tmp = 0.125 * x;
	} else if (y <= -4.2e-273) {
		tmp = t;
	} else if (y <= 6.8e-274) {
		tmp = 0.125 * x;
	} else if (y <= 1.85e-75) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = y * (z * (-0.5d0))
    if (y <= (-2.05d+42)) then
        tmp = t_1
    else if (y <= (-1.3d-34)) then
        tmp = 0.125d0 * x
    else if (y <= (-4.2d-273)) then
        tmp = t
    else if (y <= 6.8d-274) then
        tmp = 0.125d0 * x
    else if (y <= 1.85d-75) then
        tmp = t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if (y <= -2.05e+42) {
		tmp = t_1;
	} else if (y <= -1.3e-34) {
		tmp = 0.125 * x;
	} else if (y <= -4.2e-273) {
		tmp = t;
	} else if (y <= 6.8e-274) {
		tmp = 0.125 * x;
	} else if (y <= 1.85e-75) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (z * -0.5)
	tmp = 0
	if y <= -2.05e+42:
		tmp = t_1
	elif y <= -1.3e-34:
		tmp = 0.125 * x
	elif y <= -4.2e-273:
		tmp = t
	elif y <= 6.8e-274:
		tmp = 0.125 * x
	elif y <= 1.85e-75:
		tmp = t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z * -0.5))
	tmp = 0.0
	if (y <= -2.05e+42)
		tmp = t_1;
	elseif (y <= -1.3e-34)
		tmp = Float64(0.125 * x);
	elseif (y <= -4.2e-273)
		tmp = t;
	elseif (y <= 6.8e-274)
		tmp = Float64(0.125 * x);
	elseif (y <= 1.85e-75)
		tmp = t;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (z * -0.5);
	tmp = 0.0;
	if (y <= -2.05e+42)
		tmp = t_1;
	elseif (y <= -1.3e-34)
		tmp = 0.125 * x;
	elseif (y <= -4.2e-273)
		tmp = t;
	elseif (y <= 6.8e-274)
		tmp = 0.125 * x;
	elseif (y <= 1.85e-75)
		tmp = t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.05e+42], t$95$1, If[LessEqual[y, -1.3e-34], N[(0.125 * x), $MachinePrecision], If[LessEqual[y, -4.2e-273], t, If[LessEqual[y, 6.8e-274], N[(0.125 * x), $MachinePrecision], If[LessEqual[y, 1.85e-75], t, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.05e42 or 1.85000000000000012e-75 < y

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. div-inv 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. 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 y around inf 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*r* 59.8%

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

   8. Simplified 59.8%

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

   if -2.05e42 < y < -1.3e-34 or -4.2000000000000004e-273 < y < 6.79999999999999962e-274

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. div-inv 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. 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 53.1%

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

   if -1.3e-34 < y < -4.2000000000000004e-273 or 6.79999999999999962e-274 < y < 1.85000000000000012e-75

   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 t around inf 49.9%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-34}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-273}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \end{array} \]

Alternative 6: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{-19} \lor \neg \left(z \cdot y \leq 2 \cdot 10^{+86}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -2e-19) (not (<= (* z y) 2e+86)))
   (- t (* (* z y) 0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -2e-19) || !((z * y) <= 2e+86)) {
		tmp = t - ((z * y) * 0.5);
	} 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) <= (-2d-19)) .or. (.not. ((z * y) <= 2d+86))) then
        tmp = t - ((z * y) * 0.5d0)
    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) <= -2e-19) || !((z * y) <= 2e+86)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -2e-19) or not ((z * y) <= 2e+86):
		tmp = t - ((z * y) * 0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -2e-19) || !(Float64(z * y) <= 2e+86))
		tmp = Float64(t - Float64(Float64(z * y) * 0.5));
	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) <= -2e-19) || ~(((z * y) <= 2e+86)))
		tmp = t - ((z * y) * 0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -2e-19], N[Not[LessEqual[N[(z * y), $MachinePrecision], 2e+86]], $MachinePrecision]], N[(t - N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2e-19 or 2e86 < (*.f64 y z)

   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 x around 0 88.5%

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

   if -2e-19 < (*.f64 y z) < 2e86

   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 92.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -2 \cdot 10^{-19} \lor \neg \left(z \cdot y \leq 2 \cdot 10^{+86}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

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 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. Final simplification 100.0%

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

Alternative 8: 50.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+28}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+73}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.8e+28) t (if (<= t 1.7e+73) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.8e+28) {
		tmp = t;
	} else if (t <= 1.7e+73) {
		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 <= (-5.8d+28)) then
        tmp = t
    else if (t <= 1.7d+73) 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 <= -5.8e+28) {
		tmp = t;
	} else if (t <= 1.7e+73) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.8e+28:
		tmp = t
	elif t <= 1.7e+73:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.8e+28)
		tmp = t;
	elseif (t <= 1.7e+73)
		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 <= -5.8e+28)
		tmp = t;
	elseif (t <= 1.7e+73)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.8e+28], t, If[LessEqual[t, 1.7e+73], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8000000000000002e28 or 1.7000000000000001e73 < t

   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 t around inf 62.8%

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

   if -5.8000000000000002e28 < t < 1.7000000000000001e73

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. div-inv 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. 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 46.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+28}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+73}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9: 33.2% 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 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 t around inf 32.1%

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

5. Final simplification 32.1%

   \[\leadsto t \]

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 2023181 
(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))