Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 97.4% → 97.0%

Time: 10.4s

Alternatives: 17

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x - y) / (z - y)) * 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 = ((x - y) / (z - y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x - y) / (z - y)) * 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 = ((x - y) / (z - y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 40000:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{z - y}{t\_m}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 40000.0)
    (/ (* t_m (- x y)) (- z y))
    (/ (- x y) (/ (- z y) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 40000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) / ((z - y) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 40000.0d0) then
        tmp = (t_m * (x - y)) / (z - y)
    else
        tmp = (x - y) / ((z - y) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 40000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) / ((z - y) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 40000.0:
		tmp = (t_m * (x - y)) / (z - y)
	else:
		tmp = (x - y) / ((z - y) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 40000.0)
		tmp = Float64(Float64(t_m * Float64(x - y)) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) / Float64(Float64(z - y) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 40000.0)
		tmp = (t_m * (x - y)) / (z - y);
	else
		tmp = (x - y) / ((z - y) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 40000.0], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4e4

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 4e4 < t

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 40000:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{z - y}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.0002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ x (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.0002)
      t_2
      (if (<= t_3 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_3 2e+192) t_2 (* x (/ t_m (- z y))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (x / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0002) {
		tmp = t_2;
	} else if (t_3 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 2e+192) {
		tmp = t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(x / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.0002)
		tmp = t_2;
	elseif (t_3 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 2e+192)
		tmp = t_2;
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.0002], t$95$2, If[LessEqual[t$95$3, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2e+192], t$95$2, N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000001e-4 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000008e192

   1. Initial program 98.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.1

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

   7. Applied rewrites 88.1%

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

   9. Applied rewrites 83.4%

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

   11. Applied rewrites 95.5%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.8%

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

   if 2.00000000000000008e192 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.0002:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -50.0)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* t_m (/ (- x y) z))
        (if (<= t_2 20.0)
          (fma t_m (/ (- z x) y) t_m)
          (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -50.0], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.1

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

   7. Applied rewrites 86.1%

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

   9. Applied rewrites 85.2%

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

   11. Applied rewrites 94.4%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;t\_m \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -50.0)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* t_m (/ (- x y) z))
        (if (<= t_2 20.0) (* t_m (- 1.0 (/ x y))) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * (1.0 - (x / y));
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-50.0d0)) then
        tmp = t_m * (x / (z - y))
    else if (t_2 <= 0.4d0) then
        tmp = t_m * ((x - y) / z)
    else if (t_2 <= 20.0d0) then
        tmp = t_m * (1.0d0 - (x / y))
    else
        tmp = (t_m * x) / (z - y)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * (1.0 - (x / y));
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -50.0:
		tmp = t_m * (x / (z - y))
	elif t_2 <= 0.4:
		tmp = t_m * ((x - y) / z)
	elif t_2 <= 20.0:
		tmp = t_m * (1.0 - (x / y))
	else:
		tmp = (t_m * x) / (z - y)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_2 <= 20.0)
		tmp = Float64(t_m * Float64(1.0 - Float64(x / y)));
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -50.0)
		tmp = t_m * (x / (z - y));
	elseif (t_2 <= 0.4)
		tmp = t_m * ((x - y) / z);
	elseif (t_2 <= 20.0)
		tmp = t_m * (1.0 - (x / y));
	else
		tmp = (t_m * x) / (z - y);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -50.0], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.1

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

   7. Applied rewrites 86.1%

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

   9. Applied rewrites 85.2%

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

   11. Applied rewrites 94.4%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.0002:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;t\_m \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -0.0002)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_2 20.0) (* t_m (- 1.0 (/ x y))) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.0002) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * (1.0 - (x / y));
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-0.0002d0)) then
        tmp = t_m * (x / (z - y))
    else if (t_2 <= 0.4d0) then
        tmp = (x - y) * (t_m / z)
    else if (t_2 <= 20.0d0) then
        tmp = t_m * (1.0d0 - (x / y))
    else
        tmp = (t_m * x) / (z - y)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.0002) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * (1.0 - (x / y));
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -0.0002:
		tmp = t_m * (x / (z - y))
	elif t_2 <= 0.4:
		tmp = (x - y) * (t_m / z)
	elif t_2 <= 20.0:
		tmp = t_m * (1.0 - (x / y))
	else:
		tmp = (t_m * x) / (z - y)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -0.0002)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 20.0)
		tmp = Float64(t_m * Float64(1.0 - Float64(x / y)));
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -0.0002)
		tmp = t_m * (x / (z - y));
	elseif (t_2 <= 0.4)
		tmp = (x - y) * (t_m / z);
	elseif (t_2 <= 20.0)
		tmp = t_m * (1.0 - (x / y));
	else
		tmp = (t_m * x) / (z - y);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -0.0002], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000001e-4

   1. Initial program 96.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.4

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 85.6%

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

   11. Applied rewrites 94.6%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.0002:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.0002:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -0.0002)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.0002) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -0.0002)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -0.0002], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000001e-4

   1. Initial program 96.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.4

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 85.6%

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

   11. Applied rewrites 94.6%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 94.1

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

   7. Applied rewrites 94.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.0002:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \frac{t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -50:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* x (/ t_m (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -50.0)
      t_2
      (if (<= t_3 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_3 20.0) (fma t_m (/ z y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -50.0) {
		tmp = t_2;
	} else if (t_3 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(x * Float64(t_m / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -50.0)
		tmp = t_2;
	elseif (t_3 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -50.0], t$95$2, If[LessEqual[t$95$3, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50 or 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 92.0

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 87.9%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* (- x y) (/ t_m (- z y)))
      (if (<= t_2 20.0) (fma t_m (/ (- z x) y) t_m) (/ (* t_m x) (- z y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / (z - y));
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{-y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* (- x y) (/ t_m z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* t_m x) (- y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / -y;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(-y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / (-y)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 94.1

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 59.8%

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

Alternative 10: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{-y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* t_m x) (- y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / -y;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(-y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / (-y)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 94.1

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 59.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{-y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 20.0) (fma t_m (/ z y) t_m) (* x (/ t_m (- y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = x * (t_m / -y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(x * Float64(t_m / Float64(-y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(x * N[(t$95$m / (-y)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 95.4

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

   7. Applied rewrites 95.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 60.5%

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

   12. Applied rewrites 57.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 20.0) (fma t_m (/ z y) t_m) (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 20.0) (* t_m 1.0) (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 0.4d0) then
        tmp = t_m * (x / z)
    else if (t_2 <= 20.0d0) then
        tmp = t_m * 1.0d0
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 0.4:
		tmp = t_m * (x / z)
	elif t_2 <= 20.0:
		tmp = t_m * 1.0
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = Float64(t_m * 1.0);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 0.4)
		tmp = t_m * (x / z);
	elseif (t_2 <= 20.0)
		tmp = t_m * 1.0;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * 1.0), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{1} \cdot t \]

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot x}{z}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 20:\\ \;\;\;\;t\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (* t_m x) z)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 0.2) t_2 (if (<= t_3 20.0) (* t_m 1.0) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m * x) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.2) {
		tmp = t_2;
	} else if (t_3 <= 20.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m * x) / z
    t_3 = (x - y) / (z - y)
    if (t_3 <= 0.2d0) then
        tmp = t_2
    else if (t_3 <= 20.0d0) then
        tmp = t_m * 1.0d0
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m * x) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.2) {
		tmp = t_2;
	} else if (t_3 <= 20.0) {
		tmp = t_m * 1.0;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m * x) / z
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 0.2:
		tmp = t_2
	elif t_3 <= 20.0:
		tmp = t_m * 1.0
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m * x) / z)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 0.2)
		tmp = t_2;
	elseif (t_3 <= 20.0)
		tmp = Float64(t_m * 1.0);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m * x) / z;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 0.2)
		tmp = t_2;
	elseif (t_3 <= 20.0)
		tmp = t_m * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 0.2], t$95$2, If[LessEqual[t$95$3, 20.0], N[(t$95$m * 1.0), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001 or 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{1} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.2:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_m \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (* t_s (if (<= t_2 2e+192) (* t_m t_2) (* x (/ t_m (- z y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e+192) {
		tmp = t_m * t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 2d+192) then
        tmp = t_m * t_2
    else
        tmp = x * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e+192) {
		tmp = t_m * t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 2e+192:
		tmp = t_m * t_2
	else:
		tmp = x * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 2e+192)
		tmp = Float64(t_m * t_2);
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 2e+192)
		tmp = t_m * t_2;
	else
		tmp = x * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 2e+192], N[(t$95$m * t$95$2), $MachinePrecision], N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000008e192

   1. Initial program 98.2%

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

   if 2.00000000000000008e192 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t \cdot \frac{x - y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 20000000000000:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 20000000000000.0)
    (/ (* t_m (- x y)) (- z y))
    (* (- x y) (/ t_m (- z y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 20000000000000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 20000000000000.0d0) then
        tmp = (t_m * (x - y)) / (z - y)
    else
        tmp = (x - y) * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 20000000000000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 20000000000000.0:
		tmp = (t_m * (x - y)) / (z - y)
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 20000000000000.0)
		tmp = Float64(Float64(t_m * Float64(x - y)) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 20000000000000.0)
		tmp = (t_m * (x - y)) / (z - y);
	else
		tmp = (x - y) * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 20000000000000.0], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2e13

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 2e13 < t

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 20000000000000:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \cdot t \]
4. Step-by-step derivation

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{1} \cdot t \]

6. Final simplification 38.1%

   \[\leadsto t \cdot 1 \]
7. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return t / ((z - y) / (x - 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 = t / ((z - y) / (x - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

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

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