Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 83.5% → 96.8%

Time: 10.1s

Alternatives: 11

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x\_m, x\_m \cdot y\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e+32)
    (/ (fma (- z) x_m (* x_m y)) (- t z))
    (* x_m (/ (- y z) (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+32) {
		tmp = fma(-z, x_m, (x_m * y)) / (t - z);
	} else {
		tmp = x_m * ((y - z) / (t - z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e+32)
		tmp = Float64(fma(Float64(-z), x_m, Float64(x_m * y)) / Float64(t - z));
	else
		tmp = Float64(x_m * Float64(Float64(y - z) / Float64(t - z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+32], N[(N[((-z) * x$95$m + N[(x$95$m * y), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000011e32

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 2.00000000000000011e32 < x

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x, x \cdot y\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (- y z)) (- t z)) -5e-28)
    (* (- y z) (/ x_m (- t z)))
    (* x_m (/ (- y z) (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((x_m * (y - z)) / (t - z)) <= -5e-28) {
		tmp = (y - z) * (x_m / (t - z));
	} else {
		tmp = x_m * ((y - z) / (t - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x_m * (y - z)) / (t - z)) <= (-5d-28)) then
        tmp = (y - z) * (x_m / (t - z))
    else
        tmp = x_m * ((y - z) / (t - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((x_m * (y - z)) / (t - z)) <= -5e-28) {
		tmp = (y - z) * (x_m / (t - z));
	} else {
		tmp = x_m * ((y - z) / (t - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if ((x_m * (y - z)) / (t - z)) <= -5e-28:
		tmp = (y - z) * (x_m / (t - z))
	else:
		tmp = x_m * ((y - z) / (t - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y - z)) / Float64(t - z)) <= -5e-28)
		tmp = Float64(Float64(y - z) * Float64(x_m / Float64(t - z)));
	else
		tmp = Float64(x_m * Float64(Float64(y - z) / Float64(t - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (((x_m * (y - z)) / (t - z)) <= -5e-28)
		tmp = (y - z) * (x_m / (t - z));
	else
		tmp = x_m * ((y - z) / (t - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], -5e-28], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.0000000000000002e-28

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -5.0000000000000002e-28 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{-28}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{z}{z - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ z (- z t)))))
   (*
    x_s
    (if (<= z -3.6e+122)
      t_1
      (if (<= z 1.5e+146) (* (- y z) (/ x_m (- t z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (z / (z - t));
	double tmp;
	if (z <= -3.6e+122) {
		tmp = t_1;
	} else if (z <= 1.5e+146) {
		tmp = (y - z) * (x_m / (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (z / (z - t))
    if (z <= (-3.6d+122)) then
        tmp = t_1
    else if (z <= 1.5d+146) then
        tmp = (y - z) * (x_m / (t - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (z / (z - t));
	double tmp;
	if (z <= -3.6e+122) {
		tmp = t_1;
	} else if (z <= 1.5e+146) {
		tmp = (y - z) * (x_m / (t - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (z / (z - t))
	tmp = 0
	if z <= -3.6e+122:
		tmp = t_1
	elif z <= 1.5e+146:
		tmp = (y - z) * (x_m / (t - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(z / Float64(z - t)))
	tmp = 0.0
	if (z <= -3.6e+122)
		tmp = t_1;
	elseif (z <= 1.5e+146)
		tmp = Float64(Float64(y - z) * Float64(x_m / Float64(t - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (z / (z - t));
	tmp = 0.0;
	if (z <= -3.6e+122)
		tmp = t_1;
	elseif (z <= 1.5e+146)
		tmp = (y - z) * (x_m / (t - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.6e+122], t$95$1, If[LessEqual[z, 1.5e+146], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6000000000000003e122 or 1.50000000000000001e146 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 90.4

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

   8. Applied rewrites 90.4%

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

   if -3.6000000000000003e122 < z < 1.50000000000000001e146

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-33}:\\ \;\;\;\;x\_m \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-70}:\\ \;\;\;\;\frac{x\_m \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -6.8e-33)
    (* x_m (/ z (- z t)))
    (if (<= z 1.28e-70) (/ (* x_m y) (- t z)) (fma (/ y (- z)) x_m x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -6.8e-33) {
		tmp = x_m * (z / (z - t));
	} else if (z <= 1.28e-70) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = fma((y / -z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -6.8e-33)
		tmp = Float64(x_m * Float64(z / Float64(z - t)));
	elseif (z <= 1.28e-70)
		tmp = Float64(Float64(x_m * y) / Float64(t - z));
	else
		tmp = fma(Float64(y / Float64(-z)), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -6.8e-33], N[(x$95$m * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.28e-70], N[(N[(x$95$m * y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(y / (-z)), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -6.8000000000000001e-33

   1. Initial program 77.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 13.5

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

   5. Applied rewrites 13.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -6.8000000000000001e-33 < z < 1.28e-70

   1. Initial program 94.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 1.28e-70 < z

   1. Initial program 80.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 74.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+46}:\\ \;\;\;\;x\_m \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ y (- t z)))))
   (*
    x_s
    (if (<= y -1.2e+36) t_1 (if (<= y 2e+46) (* x_m (/ z (- z t))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (y / (t - z));
	double tmp;
	if (y <= -1.2e+36) {
		tmp = t_1;
	} else if (y <= 2e+46) {
		tmp = x_m * (z / (z - t));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (y / (t - z))
    if (y <= (-1.2d+36)) then
        tmp = t_1
    else if (y <= 2d+46) then
        tmp = x_m * (z / (z - t))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (y / (t - z));
	double tmp;
	if (y <= -1.2e+36) {
		tmp = t_1;
	} else if (y <= 2e+46) {
		tmp = x_m * (z / (z - t));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (y / (t - z))
	tmp = 0
	if y <= -1.2e+36:
		tmp = t_1
	elif y <= 2e+46:
		tmp = x_m * (z / (z - t))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(y / Float64(t - z)))
	tmp = 0.0
	if (y <= -1.2e+36)
		tmp = t_1;
	elseif (y <= 2e+46)
		tmp = Float64(x_m * Float64(z / Float64(z - t)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (y / (t - z));
	tmp = 0.0;
	if (y <= -1.2e+36)
		tmp = t_1;
	elseif (y <= 2e+46)
		tmp = x_m * (z / (z - t));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.2e+36], t$95$1, If[LessEqual[y, 2e+46], N[(x$95$m * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999996e36 or 2e46 < y

   1. Initial program 83.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -1.19999999999999996e36 < y < 2e46

   1. Initial program 87.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 6: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{t}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-53}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.05e+95)
    (fma x_m (/ t z) x_m)
    (if (<= z 1.75e-53) (* x_m (/ y (- t z))) (* x_m 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+95) {
		tmp = fma(x_m, (t / z), x_m);
	} else if (z <= 1.75e-53) {
		tmp = x_m * (y / (t - z));
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.05e+95)
		tmp = fma(x_m, Float64(t / z), x_m);
	elseif (z <= 1.75e-53)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	else
		tmp = Float64(x_m * 1.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.05e+95], N[(x$95$m * N[(t / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 1.75e-53], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e95

   1. Initial program 69.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 89.3

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 75.6%

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

   if -1.05e95 < z < 1.74999999999999997e-53

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if 1.74999999999999997e-53 < z

   1. Initial program 80.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.2

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 74.5

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

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 60.5%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{t}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.3e+51)
    (fma x_m (/ t z) x_m)
    (if (<= z 6.6e-54) (* x_m (/ y t)) (* x_m 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e+51) {
		tmp = fma(x_m, (t / z), x_m);
	} else if (z <= 6.6e-54) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.3e+51)
		tmp = fma(x_m, Float64(t / z), x_m);
	elseif (z <= 6.6e-54)
		tmp = Float64(x_m * Float64(y / t));
	else
		tmp = Float64(x_m * 1.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.3e+51], N[(x$95$m * N[(t / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 6.6e-54], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.2999999999999997e51

   1. Initial program 73.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 8.0

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

   5. Applied rewrites 8.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 67.7%

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

   if -3.2999999999999997e51 < z < 6.59999999999999986e-54

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 65.8%

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

   if 6.59999999999999986e-54 < z

   1. Initial program 80.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.2

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 74.5

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

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 60.5%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.1e+51)
    (* x_m 1.0)
    (if (<= z 6.6e-54) (* x_m (/ y t)) (* x_m 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+51) {
		tmp = x_m * 1.0;
	} else if (z <= 6.6e-54) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.1d+51)) then
        tmp = x_m * 1.0d0
    else if (z <= 6.6d-54) then
        tmp = x_m * (y / t)
    else
        tmp = x_m * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+51) {
		tmp = x_m * 1.0;
	} else if (z <= 6.6e-54) {
		tmp = x_m * (y / t);
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.1e+51:
		tmp = x_m * 1.0
	elif z <= 6.6e-54:
		tmp = x_m * (y / t)
	else:
		tmp = x_m * 1.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.1e+51)
		tmp = Float64(x_m * 1.0);
	elseif (z <= 6.6e-54)
		tmp = Float64(x_m * Float64(y / t));
	else
		tmp = Float64(x_m * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.1e+51)
		tmp = x_m * 1.0;
	elseif (z <= 6.6e-54)
		tmp = x_m * (y / t);
	else
		tmp = x_m * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.1e+51], N[(x$95$m * 1.0), $MachinePrecision], If[LessEqual[z, 6.6e-54], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000011e51 or 6.59999999999999986e-54 < z

   1. Initial program 78.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 13.1

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 63.2%

       \[\leadsto x \cdot 1 \]

   if -3.10000000000000011e51 < z < 6.59999999999999986e-54

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 65.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.1e+51)
    (* x_m 1.0)
    (if (<= z 6.6e-54) (* y (/ x_m t)) (* x_m 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+51) {
		tmp = x_m * 1.0;
	} else if (z <= 6.6e-54) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.1d+51)) then
        tmp = x_m * 1.0d0
    else if (z <= 6.6d-54) then
        tmp = y * (x_m / t)
    else
        tmp = x_m * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+51) {
		tmp = x_m * 1.0;
	} else if (z <= 6.6e-54) {
		tmp = y * (x_m / t);
	} else {
		tmp = x_m * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.1e+51:
		tmp = x_m * 1.0
	elif z <= 6.6e-54:
		tmp = y * (x_m / t)
	else:
		tmp = x_m * 1.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.1e+51)
		tmp = Float64(x_m * 1.0);
	elseif (z <= 6.6e-54)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = Float64(x_m * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.1e+51)
		tmp = x_m * 1.0;
	elseif (z <= 6.6e-54)
		tmp = y * (x_m / t);
	else
		tmp = x_m * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.1e+51], N[(x$95$m * 1.0), $MachinePrecision], If[LessEqual[z, 6.6e-54], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000011e51 or 6.59999999999999986e-54 < z

   1. Initial program 78.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 13.1

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 63.2%

       \[\leadsto x \cdot 1 \]

   if -3.10000000000000011e51 < z < 6.59999999999999986e-54

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 65.2%

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

Alternative 10: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e+32) (/ (* x_m (- y z)) (- t z)) (* x_m (/ (- y z) (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+32) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = x_m * ((y - z) / (t - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2d+32) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = x_m * ((y - z) / (t - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2e+32) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = x_m * ((y - z) / (t - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2e+32:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = x_m * ((y - z) / (t - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2e+32)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(x_m * Float64(Float64(y - z) / Float64(t - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2e+32)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = x_m * ((y - z) / (t - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+32], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000011e32

   1. Initial program 91.8%

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

   if 2.00000000000000011e32 < x

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot 1\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* x_m 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * 1.0);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m * 1.0d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * 1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m * 1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * 1.0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m * 1.0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 39.5

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

5. Applied rewrites 39.5%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. remove-double-neg N/A

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

   9. sub-neg N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 53.0

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

8. Applied rewrites 53.0%

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

9. Taylor expanded in z around inf

   \[\leadsto x \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 34.5%

    \[\leadsto x \cdot 1 \]
12. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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