Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 67.9% → 89.2%

Time: 11.7s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ (- y a) z) t)))
   (if (<= z -1.5e+123)
     t_1
     (if (<= z 1.9e+189) (fma (- t x) (/ (- y z) (- a z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), ((y - a) / z), t);
	double tmp;
	if (z <= -1.5e+123) {
		tmp = t_1;
	} else if (z <= 1.9e+189) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(Float64(y - a) / z), t)
	tmp = 0.0
	if (z <= -1.5e+123)
		tmp = t_1;
	elseif (z <= 1.9e+189)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.5e+123], t$95$1, If[LessEqual[z, 1.9e+189], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000004e123 or 1.8999999999999999e189 < z

   1. Initial program 30.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 94.8%

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

   if -1.50000000000000004e123 < z < 1.8999999999999999e189

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.4

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

   4. Applied egg-rr 92.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -2.8e-14)
     t_1
     (if (<= z 3.2e-171)
       (fma (- y z) (/ (- t x) a) x)
       (if (<= z 1.9e+189) (fma (/ (- y z) (- a z)) t x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 3.2e-171) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else if (z <= 1.9e+189) {
		tmp = fma(((y - z) / (a - z)), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 3.2e-171)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	elseif (z <= 1.9e+189)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 3.2e-171], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.9e+189], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8000000000000001e-14 or 1.8999999999999999e189 < z

   1. Initial program 45.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 90.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 84.4

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

   8. Simplified 84.4%

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

   if -2.8000000000000001e-14 < z < 3.2000000000000001e-171

   1. Initial program 88.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 89.1

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

   5. Simplified 89.1%

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

   if 3.2000000000000001e-171 < z < 1.8999999999999999e189

   1. Initial program 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 75.6

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

   4. Applied egg-rr 75.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 62.6

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

   7. Simplified 62.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 73.3

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

   9. Applied egg-rr 73.3%

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

Alternative 3: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ (- y a) z) t)))
   (if (<= z -2.8e-14)
     t_1
     (if (<= z 1.4e-16) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), ((y - a) / z), t);
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 1.4e-16) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(Float64(y - a) / z), t)
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 1.4e-16)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 1.4e-16], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-14 or 1.4000000000000001e-16 < z

   1. Initial program 51.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.5%

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

   if -2.8000000000000001e-14 < z < 1.4000000000000001e-16

   1. Initial program 89.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 85.3

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

   5. Simplified 85.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -2.8e-14)
     t_1
     (if (<= z 1.4e-16) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 1.4e-16) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 1.4e-16)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 1.4e-16], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-14 or 1.4000000000000001e-16 < z

   1. Initial program 51.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.8

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

   8. Simplified 76.8%

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

   if -2.8000000000000001e-14 < z < 1.4000000000000001e-16

   1. Initial program 89.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 85.3

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

   5. Simplified 85.3%

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

Alternative 5: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -2.8e-14)
     t_1
     (if (<= z 1.4e-16) (fma (- t x) (/ (- y z) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 1.4e-16) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 1.4e-16)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 1.4e-16], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-14 or 1.4000000000000001e-16 < z

   1. Initial program 51.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.8

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

   8. Simplified 76.8%

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

   if -2.8000000000000001e-14 < z < 1.4000000000000001e-16

   1. Initial program 89.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 95.6

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.7

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

   7. Simplified 83.7%

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

Alternative 6: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -2.9e-15) t_1 (if (<= z 2.4e-27) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -2.9e-15) {
		tmp = t_1;
	} else if (z <= 2.4e-27) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -2.9e-15)
		tmp = t_1;
	elseif (z <= 2.4e-27)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.9e-15], t$95$1, If[LessEqual[z, 2.4e-27], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.90000000000000019e-15 or 2.40000000000000002e-27 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.6

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

   8. Simplified 76.6%

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

   if -2.90000000000000019e-15 < z < 2.40000000000000002e-27

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.3

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

   5. Simplified 81.3%

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

Alternative 7: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - y \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* y (/ t z)))))
   (if (<= z -2.7e-13) t_1 (if (<= z 1.35e+37) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (y * (t / z));
	double tmp;
	if (z <= -2.7e-13) {
		tmp = t_1;
	} else if (z <= 1.35e+37) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(y * Float64(t / z)))
	tmp = 0.0
	if (z <= -2.7e-13)
		tmp = t_1;
	elseif (z <= 1.35e+37)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.7e-13], t$95$1, If[LessEqual[z, 1.35e+37], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000011e-13 or 1.34999999999999993e37 < z

   1. Initial program 49.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 78.3

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

   8. Simplified 78.3%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 62.7

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

   11. Simplified 62.7%

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

   if -2.70000000000000011e-13 < z < 1.34999999999999993e37

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.1

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

   5. Simplified 77.1%

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

Alternative 8: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - y \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* y (/ t z)))))
   (if (<= z -2.3e-13) t_1 (if (<= z 3.6e+36) (fma t (/ (- y z) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (y * (t / z));
	double tmp;
	if (z <= -2.3e-13) {
		tmp = t_1;
	} else if (z <= 3.6e+36) {
		tmp = fma(t, ((y - z) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(y * Float64(t / z)))
	tmp = 0.0
	if (z <= -2.3e-13)
		tmp = t_1;
	elseif (z <= 3.6e+36)
		tmp = fma(t, Float64(Float64(y - z) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.3e-13], t$95$1, If[LessEqual[z, 3.6e+36], N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999979e-13 or 3.5999999999999997e36 < z

   1. Initial program 49.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 78.3

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

   8. Simplified 78.3%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 62.7

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

   11. Simplified 62.7%

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

   if -2.29999999999999979e-13 < z < 3.5999999999999997e36

   1. Initial program 89.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 89.3

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 66.8

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

   7. Simplified 66.8%

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

   8. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 62.1

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

   10. Simplified 62.1%

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

Alternative 9: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - y \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* y (/ t z)))))
   (if (<= z -2.8e-14) t_1 (if (<= z 1.2e-27) (fma t (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (y * (t / z));
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 1.2e-27) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(y * Float64(t / z)))
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 1.2e-27)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 1.2e-27], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-14 or 1.20000000000000001e-27 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.6

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

   8. Simplified 76.6%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 60.3

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

   11. Simplified 60.3%

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

   if -2.8000000000000001e-14 < z < 1.20000000000000001e-27

   1. Initial program 89.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 89.3

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 67.2

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

   7. Simplified 67.2%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 59.7

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

   10. Simplified 59.7%

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

Alternative 10: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot \left(z - y\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ t z) (- z y))))
   (if (<= z -2.8e-14) t_1 (if (<= z 1.2e-27) (fma t (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t / z) * (z - y);
	double tmp;
	if (z <= -2.8e-14) {
		tmp = t_1;
	} else if (z <= 1.2e-27) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / z) * Float64(z - y))
	tmp = 0.0
	if (z <= -2.8e-14)
		tmp = t_1;
	elseif (z <= 1.2e-27)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-14], t$95$1, If[LessEqual[z, 1.2e-27], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-14 or 1.20000000000000001e-27 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.6

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

   8. Simplified 76.6%

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

   9. Taylor expanded in t around -inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. *-inverses N/A

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

       4. div-sub N/A

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

       5. distribute-lft-neg-in N/A

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

       6. associate-/l* N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. distribute-lft-neg-in N/A

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

       10. mul-1-neg N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. mul-1-neg N/A

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

       14. +-commutative N/A

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

       15. distribute-lft-in N/A

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

       16. neg-mul-1 N/A

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

       17. mul-1-neg N/A

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

       18. remove-double-neg N/A

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

       19. mul-1-neg N/A

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

       20. unsub-neg N/A

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

       21. lower--.f64 N/A

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

       22. lower-/.f64 55.4

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

   11. Simplified 55.4%

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

   if -2.8000000000000001e-14 < z < 1.20000000000000001e-27

   1. Initial program 89.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 89.3

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 67.2

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

   7. Simplified 67.2%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 59.7

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

   10. Simplified 59.7%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-13}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.8e-13)
   (+ x t)
   (if (<= z 5.5e+95) (fma t (/ y a) x) (+ t (- x x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.8e-13) {
		tmp = x + t;
	} else if (z <= 5.5e+95) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t + (x - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.8e-13)
		tmp = Float64(x + t);
	elseif (z <= 5.5e+95)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(t + Float64(x - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.8e-13], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.5e+95], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(x - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.79999999999999986e-13

   1. Initial program 53.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 53.4

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 48.2

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

   7. Simplified 48.2%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t + x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 46.4

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

   10. Simplified 46.4%

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

   if -8.79999999999999986e-13 < z < 5.4999999999999997e95

   1. Initial program 88.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 88.1

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 66.3

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

   7. Simplified 66.3%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 55.1

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

   10. Simplified 55.1%

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

   if 5.4999999999999997e95 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 39.1

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

   5. Simplified 39.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \left(x - x\right)} \]

      6. lower--.f64 57.2

         \[\leadsto t - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 57.2%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-13}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+96}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.7e+108) (/ (* t y) a) (if (<= y 3.9e+96) (+ x t) (* x (/ y z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.7e+108) {
		tmp = (t * y) / a;
	} else if (y <= 3.9e+96) {
		tmp = x + t;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-1.7d+108)) then
        tmp = (t * y) / a
    else if (y <= 3.9d+96) then
        tmp = x + t
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.7e+108) {
		tmp = (t * y) / a;
	} else if (y <= 3.9e+96) {
		tmp = x + t;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.7e+108:
		tmp = (t * y) / a
	elif y <= 3.9e+96:
		tmp = x + t
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.7e+108)
		tmp = Float64(Float64(t * y) / a);
	elseif (y <= 3.9e+96)
		tmp = Float64(x + t);
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.7e+108)
		tmp = (t * y) / a;
	elseif (y <= 3.9e+96)
		tmp = x + t;
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.7e+108], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 3.9e+96], N[(x + t), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.69999999999999998e108

   1. Initial program 81.1%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 74.2

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

   5. Simplified 74.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 46.4

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

   8. Simplified 46.4%

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

   9. Taylor expanded in t around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 32.4

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

   11. Simplified 32.4%

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

   if -1.69999999999999998e108 < y < 3.9e96

   1. Initial program 68.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 63.1

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

   7. Simplified 63.1%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t + x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 49.6

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

   10. Simplified 49.6%

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

   if 3.9e96 < y

   1. Initial program 59.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 55.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 56.1

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

   8. Simplified 56.1%

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

   9. Taylor expanded in x around inf

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 40.0

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

   11. Simplified 40.0%

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

Alternative 13: 39.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+96}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ y z))))
   (if (<= y -4.7e+111) t_1 (if (<= y 3.9e+96) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / z);
	double tmp;
	if (y <= -4.7e+111) {
		tmp = t_1;
	} else if (y <= 3.9e+96) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y / z)
    if (y <= (-4.7d+111)) then
        tmp = t_1
    else if (y <= 3.9d+96) then
        tmp = x + t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / z);
	double tmp;
	if (y <= -4.7e+111) {
		tmp = t_1;
	} else if (y <= 3.9e+96) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * (y / z)
	tmp = 0
	if y <= -4.7e+111:
		tmp = t_1
	elif y <= 3.9e+96:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (y <= -4.7e+111)
		tmp = t_1;
	elseif (y <= 3.9e+96)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y / z);
	tmp = 0.0;
	if (y <= -4.7e+111)
		tmp = t_1;
	elseif (y <= 3.9e+96)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e+111], t$95$1, If[LessEqual[y, 3.9e+96], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.70000000000000008e111 or 3.9e96 < y

   1. Initial program 70.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 60.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 60.5

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

   8. Simplified 60.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 33.7

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

   11. Simplified 33.7%

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

   if -4.70000000000000008e111 < y < 3.9e96

   1. Initial program 69.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 69.1

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

   4. Applied egg-rr 69.1%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 63.3

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

   7. Simplified 63.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t + x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 49.3

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

   10. Simplified 49.3%

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

Alternative 14: 35.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+190}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.9e+190) (+ x t) (+ t (- x x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.9e+190) {
		tmp = x + t;
	} else {
		tmp = t + (x - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 1.9d+190) then
        tmp = x + t
    else
        tmp = t + (x - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.9e+190) {
		tmp = x + t;
	} else {
		tmp = t + (x - x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 1.9e+190:
		tmp = x + t
	else:
		tmp = t + (x - x)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.9e+190)
		tmp = Float64(x + t);
	else
		tmp = Float64(t + Float64(x - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 1.9e+190)
		tmp = x + t;
	else
		tmp = t + (x - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.9e+190], N[(x + t), $MachinePrecision], N[(t + N[(x - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.89999999999999982e190

   1. Initial program 74.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 74.3

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 59.9

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

   7. Simplified 59.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t + x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 36.9

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

   10. Simplified 36.9%

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

   if 1.89999999999999982e190 < z

   1. Initial program 21.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 41.8

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

   5. Simplified 41.8%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

         \[\leadsto \color{blue}{t - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{t - \left(x - x\right)} \]

      6. lower--.f64 71.3

         \[\leadsto t - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 71.3%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{+190}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.5% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return Float64(x + t)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + t), $MachinePrecision]

Derivation

1. Initial program 69.6%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 69.6

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

4. Applied egg-rr 69.6%

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

5. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 56.5

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

7. Simplified 56.5%

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

8. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t + x} \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-+.f64 37.8

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

10. Simplified 37.8%

    \[\leadsto \color{blue}{x + t} \]
11. Add Preprocessing

Alternative 16: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 69.6%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 22.6

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

5. Simplified 22.6%

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

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 2.8

      \[\leadsto x + \color{blue}{\left(-x\right)} \]

8. Simplified 2.8%

   \[\leadsto x + \color{blue}{\left(-x\right)} \]
9. Step-by-step derivation

   1. unsub-neg N/A

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

   2. +-inverses 2.8

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 2.8%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024212 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

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