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

Percentage Accurate: 67.9% → 90.2%

Time: 9.9s

Alternatives: 15

Speedup: 0.8×

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: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ t_2 := x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- z y) (- z a)) x))
        (t_2 (- x (/ (* (- z y) (- x t)) (- z a)))))
   (if (<= t_2 -1e-267)
     t_1
     (if (<= t_2 0.0) (- t (/ (* (- x t) (- a y)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((z - y) / (z - a)), x);
	double t_2 = x - (((z - y) * (x - t)) / (z - a));
	double tmp;
	if (t_2 <= -1e-267) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t - (((x - t) * (a - y)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(z - y) / Float64(z - a)), x)
	t_2 = Float64(x - Float64(Float64(Float64(z - y) * Float64(x - t)) / Float64(z - a)))
	tmp = 0.0
	if (t_2 <= -1e-267)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(t - Float64(Float64(Float64(x - t) * Float64(a - y)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(z - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-267], t$95$1, If[LessEqual[t$95$2, 0.0], N[(t - N[(N[(N[(x - t), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -9.9999999999999998e-268 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 75.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if -9.9999999999999998e-268 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 4.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 4.0

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

   4. Applied rewrites 4.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 4.0

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

   6. Applied rewrites 4.0%

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

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   8. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a} \leq -1 \cdot 10^{-267}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \mathbf{elif}\;x - \frac{\left(z - y\right) \cdot \left(x - t\right)}{z - a} \leq 0:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -255:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-99}:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -255.0)
     t_1
     (if (<= a 3.1e-99)
       (- t (/ (* (- x t) (- a y)) z))
       (if (<= a 3.5e+69) (* (- z y) (/ t (- z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -255.0) {
		tmp = t_1;
	} else if (a <= 3.1e-99) {
		tmp = t - (((x - t) * (a - y)) / z);
	} else if (a <= 3.5e+69) {
		tmp = (z - y) * (t / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -255.0)
		tmp = t_1;
	elseif (a <= 3.1e-99)
		tmp = Float64(t - Float64(Float64(Float64(x - t) * Float64(a - y)) / z));
	elseif (a <= 3.5e+69)
		tmp = Float64(Float64(z - y) * Float64(t / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -255.0], t$95$1, If[LessEqual[a, 3.1e-99], N[(t - N[(N[(N[(x - t), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e+69], N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -255 or 3.49999999999999987e69 < a

   1. Initial program 72.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -255 < a < 3.0999999999999999e-99

   1. Initial program 67.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 78.6

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   8. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 81.0%

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

   if 3.0999999999999999e-99 < a < 3.49999999999999987e69

   1. Initial program 67.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -255:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-99}:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -205:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -205.0)
     t_1
     (if (<= a 1.65e-91)
       (- t (/ (* (- t x) y) z))
       (if (<= a 3.5e+69) (* (- z y) (/ t (- z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -205.0) {
		tmp = t_1;
	} else if (a <= 1.65e-91) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 3.5e+69) {
		tmp = (z - y) * (t / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -205.0)
		tmp = t_1;
	elseif (a <= 1.65e-91)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 3.5e+69)
		tmp = Float64(Float64(z - y) * Float64(t / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -205.0], t$95$1, If[LessEqual[a, 1.65e-91], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e+69], N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -205 or 3.49999999999999987e69 < a

   1. Initial program 72.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -205 < a < 1.65000000000000006e-91

   1. Initial program 67.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.1%

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

   if 1.65000000000000006e-91 < a < 3.49999999999999987e69

   1. Initial program 68.9%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -205:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -95000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+117}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -95000000000.0)
     t_1
     (if (<= a 1.65e-91)
       (- t (/ (* (- t x) y) z))
       (if (<= a 1.65e+117) (* (- z y) (/ t (- z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (y / a), x);
	double tmp;
	if (a <= -95000000000.0) {
		tmp = t_1;
	} else if (a <= 1.65e-91) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 1.65e+117) {
		tmp = (z - y) * (t / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -95000000000.0)
		tmp = t_1;
	elseif (a <= 1.65e-91)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 1.65e+117)
		tmp = Float64(Float64(z - y) * Float64(t / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -95000000000.0], t$95$1, If[LessEqual[a, 1.65e-91], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e+117], N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.5e10 or 1.6499999999999999e117 < a

   1. Initial program 72.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if -9.5e10 < a < 1.65000000000000006e-91

   1. Initial program 68.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.3%

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

   if 1.65000000000000006e-91 < a < 1.6499999999999999e117

   1. Initial program 68.5%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -95000000000:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+117}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2)
   (fma (/ (fma t -1.0 x) z) (- y a) t)
   (if (<= z 0.03)
     (fma (/ (- y z) a) (- t x) x)
     (- t (* (/ (- t x) z) (- y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2) {
		tmp = fma((fma(t, -1.0, x) / z), (y - a), t);
	} else if (z <= 0.03) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = t - (((t - x) / z) * (y - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2)
		tmp = fma(Float64(fma(t, -1.0, x) / z), Float64(y - a), t);
	elseif (z <= 0.03)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2], N[(N[(N[(t * -1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 0.03], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000002

   1. Initial program 45.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 82.6%

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

   if -3.2000000000000002 < z < 0.029999999999999999

   1. Initial program 90.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if 0.029999999999999999 < z

   1. Initial program 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 79.9

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 86.9

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

   7. Applied rewrites 86.9%

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

Alternative 6: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{if}\;z \leq -3.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ (- t x) z) (- y a)))))
   (if (<= z -3.2) t_1 (if (<= z 0.03) (fma (/ (- y z) a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (((t - x) / z) * (y - a));
	double tmp;
	if (z <= -3.2) {
		tmp = t_1;
	} else if (z <= 0.03) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)))
	tmp = 0.0
	if (z <= -3.2)
		tmp = t_1;
	elseif (z <= 0.03)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2], t$95$1, If[LessEqual[z, 0.03], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000002 or 0.029999999999999999 < z

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -3.2000000000000002 < z < 0.029999999999999999

   1. Initial program 90.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

Alternative 7: 68.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -95000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.0013:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -95000000000.0)
     t_1
     (if (<= a 0.0013) (- t (/ (* (- t x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (y / a), x);
	double tmp;
	if (a <= -95000000000.0) {
		tmp = t_1;
	} else if (a <= 0.0013) {
		tmp = t - (((t - x) * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -95000000000.0)
		tmp = t_1;
	elseif (a <= 0.0013)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -95000000000.0], t$95$1, If[LessEqual[a, 0.0013], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.5e10 or 0.0012999999999999999 < a

   1. Initial program 71.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 69.5

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

   7. Applied rewrites 69.5%

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

   if -9.5e10 < a < 0.0012999999999999999

   1. Initial program 68.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.8%

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

Alternative 8: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z} \cdot t\\ \mathbf{if}\;z \leq -4400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) z) t)))
   (if (<= z -4400000.0) t_1 (if (<= z 0.03) (fma (- t x) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / z) * t;
	double tmp;
	if (z <= -4400000.0) {
		tmp = t_1;
	} else if (z <= 0.03) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / z) * t)
	tmp = 0.0
	if (z <= -4400000.0)
		tmp = t_1;
	elseif (z <= 0.03)
		tmp = fma(Float64(t - x), 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[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -4400000.0], t$95$1, If[LessEqual[z, 0.03], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e6 or 0.029999999999999999 < z

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 61.4%

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

   if -4.4e6 < z < 0.029999999999999999

   1. Initial program 90.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 71.8

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

   7. Applied rewrites 71.8%

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

Alternative 9: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z} \cdot t\\ \mathbf{if}\;z \leq -4400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) z) t)))
   (if (<= z -4400000.0) t_1 (if (<= z 3.6e-11) (fma (/ (- t x) a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / z) * t;
	double tmp;
	if (z <= -4400000.0) {
		tmp = t_1;
	} else if (z <= 3.6e-11) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / z) * t)
	tmp = 0.0
	if (z <= -4400000.0)
		tmp = t_1;
	elseif (z <= 3.6e-11)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -4400000.0], t$95$1, If[LessEqual[z, 3.6e-11], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e6 or 3.59999999999999985e-11 < z

   1. Initial program 50.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 61.1%

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

   if -4.4e6 < z < 3.59999999999999985e-11

   1. Initial program 89.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

Alternative 10: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.015:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t a) y x)))
   (if (<= a -6.8e+132) t_1 (if (<= a 0.015) (* (/ (- z y) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), y, x);
	double tmp;
	if (a <= -6.8e+132) {
		tmp = t_1;
	} else if (a <= 0.015) {
		tmp = ((z - y) / z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), y, x)
	tmp = 0.0
	if (a <= -6.8e+132)
		tmp = t_1;
	elseif (a <= 0.015)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -6.8e+132], t$95$1, If[LessEqual[a, 0.015], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.80000000000000051e132 or 0.014999999999999999 < a

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 67.3%

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

   if -6.80000000000000051e132 < a < 0.014999999999999999

   1. Initial program 69.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.3%

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

Alternative 11: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -2150000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -2150000000000.0) t_1 (if (<= z 6e+39) (fma (/ t a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -2150000000000.0) {
		tmp = t_1;
	} else if (z <= 6e+39) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -2150000000000.0)
		tmp = t_1;
	elseif (z <= 6e+39)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2150000000000.0], t$95$1, If[LessEqual[z, 6e+39], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e12 or 5.9999999999999999e39 < z

   1. Initial program 45.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 41.5

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

   5. Applied rewrites 41.5%

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

   if -2.15e12 < z < 5.9999999999999999e39

   1. Initial program 89.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 55.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2150000000000:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -88000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -88000000000.0) t_1 (if (<= z 6.5e-11) (* (/ y a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -88000000000.0) {
		tmp = t_1;
	} else if (z <= 6.5e-11) {
		tmp = (y / a) * 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 = (t - x) + x
    if (z <= (-88000000000.0d0)) then
        tmp = t_1
    else if (z <= 6.5d-11) then
        tmp = (y / a) * 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 = (t - x) + x;
	double tmp;
	if (z <= -88000000000.0) {
		tmp = t_1;
	} else if (z <= 6.5e-11) {
		tmp = (y / a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t - x) + x
	tmp = 0
	if z <= -88000000000.0:
		tmp = t_1
	elif z <= 6.5e-11:
		tmp = (y / a) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -88000000000.0)
		tmp = t_1;
	elseif (z <= 6.5e-11)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (t - x) + x;
	tmp = 0.0;
	if (z <= -88000000000.0)
		tmp = t_1;
	elseif (z <= 6.5e-11)
		tmp = (y / a) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -88000000000.0], t$95$1, If[LessEqual[z, 6.5e-11], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8e10 or 6.49999999999999953e-11 < z

   1. Initial program 50.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 40.6

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

   5. Applied rewrites 40.6%

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

   if -8.8e10 < z < 6.49999999999999953e-11

   1. Initial program 89.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 32.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -88000000000:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+32}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)))
   (if (<= y -1.55e+56) t_1 (if (<= y 1.36e+32) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / z) * y;
	double tmp;
	if (y <= -1.55e+56) {
		tmp = t_1;
	} else if (y <= 1.36e+32) {
		tmp = (t - x) + 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 = (x / z) * y
    if (y <= (-1.55d+56)) then
        tmp = t_1
    else if (y <= 1.36d+32) then
        tmp = (t - x) + 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 = (x / z) * y;
	double tmp;
	if (y <= -1.55e+56) {
		tmp = t_1;
	} else if (y <= 1.36e+32) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x / z) * y
	tmp = 0
	if y <= -1.55e+56:
		tmp = t_1
	elif y <= 1.36e+32:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / z) * y)
	tmp = 0.0
	if (y <= -1.55e+56)
		tmp = t_1;
	elseif (y <= 1.36e+32)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x / z) * y;
	tmp = 0.0;
	if (y <= -1.55e+56)
		tmp = t_1;
	elseif (y <= 1.36e+32)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.55e+56], t$95$1, If[LessEqual[y, 1.36e+32], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000002e56 or 1.3599999999999999e32 < y

   1. Initial program 68.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 33.0%

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

   if -1.55000000000000002e56 < y < 1.3599999999999999e32

   1. Initial program 70.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 31.6

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

   5. Applied rewrites 31.6%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+32}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(t - x\right) + x \end{array} \]

FPCore

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

C

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

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 = (t - x) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 23.7

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

5. Applied rewrites 23.7%

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

6. Final simplification 23.7%

   \[\leadsto \left(t - x\right) + x \]
7. Add Preprocessing

Alternative 15: 2.7% 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 69.9%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 37.9

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

5. Applied rewrites 37.9%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 2.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 84.8% 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 2024277 
(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))))