Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.5% → 92.8%

Time: 6.8s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_2 := \sqrt[3]{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-267} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{{t\_2}^{2}}, \frac{t - x}{t\_2}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))) (t_2 (cbrt (- a z))))
   (if (or (<= t_1 -1e-267) (not (<= t_1 0.0)))
     (fma (/ (- y z) (pow t_2 2.0)) (/ (- t x) t_2) x)
     (fma x (/ (- y a) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double t_2 = cbrt((a - z));
	double tmp;
	if ((t_1 <= -1e-267) || !(t_1 <= 0.0)) {
		tmp = fma(((y - z) / pow(t_2, 2.0)), ((t - x) / t_2), x);
	} else {
		tmp = fma(x, ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	t_2 = cbrt(Float64(a - z))
	tmp = 0.0
	if ((t_1 <= -1e-267) || !(t_1 <= 0.0))
		tmp = fma(Float64(Float64(y - z) / (t_2 ^ 2.0)), Float64(Float64(t - x) / t_2), x);
	else
		tmp = fma(x, Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(a - z), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-267], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(y - z), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 94.5

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

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

   1. Initial program 3.2%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-267} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -1e-267) (not (<= t_1 0.0)))
     (+ x (* (- y z) (- (/ t (- a z)) (/ x (- a z)))))
     (fma x (/ (- y a) z) t))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-267], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

   4. Applied rewrites 91.9%

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

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

   1. Initial program 3.2%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

4. Final simplification 92.8%

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

Alternative 3: 90.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -1e-267) (not (<= t_1 0.0))) t_1 (fma x (/ (- y a) z) t))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-267], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], t$95$1, N[(x * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

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

   1. Initial program 3.2%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

4. Final simplification 92.8%

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

Alternative 4: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6500000000000 \lor \neg \left(z \leq 1.62 \cdot 10^{+81}\right):\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6500000000000.0) (not (<= z 1.62e+81)))
   (fma (- (- t x)) (/ (- y a) z) t)
   (+ x (* y (/ (- t x) (- a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6500000000000.0) || !(z <= 1.62e+81)) {
		tmp = fma(-(t - x), ((y - a) / z), t);
	} else {
		tmp = x + (y * ((t - x) / (a - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6500000000000.0) || !(z <= 1.62e+81))
		tmp = fma(Float64(-Float64(t - x)), Float64(Float64(y - a) / z), t);
	else
		tmp = Float64(x + Float64(y * Float64(Float64(t - x) / Float64(a - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6500000000000.0], N[Not[LessEqual[z, 1.62e+81]], $MachinePrecision]], N[((-N[(t - x), $MachinePrecision]) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(y * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e12 or 1.62e81 < z

   1. Initial program 67.1%

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

   5. Applied rewrites 84.1%

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

   if -6.5e12 < z < 1.62e81

   1. Initial program 92.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 82.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6500000000000 \lor \neg \left(z \leq 1.62 \cdot 10^{+81}\right):\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 13000000\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 13000000.0)))
   (fma (- t x) (/ (- y z) a) x)
   (fma (- (- t x)) (/ (- y a) z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 13000000.0)) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma(-(t - x), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 13000000.0))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(-Float64(t - x)), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 13000000.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-N[(t - x), $MachinePrecision]) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 1.3e7 < a

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 90.5

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 78.8%

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

   if -0.0115 < a < 1.3e7

   1. Initial program 75.4%

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

   5. Applied rewrites 83.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 13000000\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{a}\\ \mathbf{if}\;a \leq -0.0115:\\ \;\;\;\;\mathsf{fma}\left(t - x, t\_1, x\right)\\ \mathbf{elif}\;a \leq 13000000:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\_1 \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) a)))
   (if (<= a -0.0115)
     (fma (- t x) t_1 x)
     (if (<= a 13000000.0)
       (fma (- (- t x)) (/ (- y a) z) t)
       (+ x (* t_1 (- t x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / a;
	double tmp;
	if (a <= -0.0115) {
		tmp = fma((t - x), t_1, x);
	} else if (a <= 13000000.0) {
		tmp = fma(-(t - x), ((y - a) / z), t);
	} else {
		tmp = x + (t_1 * (t - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / a)
	tmp = 0.0
	if (a <= -0.0115)
		tmp = fma(Float64(t - x), t_1, x);
	elseif (a <= 13000000.0)
		tmp = fma(Float64(-Float64(t - x)), Float64(Float64(y - a) / z), t);
	else
		tmp = Float64(x + Float64(t_1 * Float64(t - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -0.0115], N[(N[(t - x), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[a, 13000000.0], N[((-N[(t - x), $MachinePrecision]) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(t$95$1 * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0115

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 90.5

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 79.3%

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

   if -0.0115 < a < 1.3e7

   1. Initial program 75.4%

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

   5. Applied rewrites 83.9%

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

   if 1.3e7 < a

   1. Initial program 85.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 78.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 13000000:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 1.65e-6)))
   (fma (- t x) (/ (- y z) a) x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 1.65e-6)) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 1.65e-6))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 1.65e-6]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 1.65000000000000008e-6 < a

   1. Initial program 87.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 89.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 78.4%

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

   if -0.0115 < a < 1.65000000000000008e-6

   1. Initial program 75.8%

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 1.65e-6)))
   (fma (- y z) (/ (- t x) a) x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 1.65e-6)) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 1.65e-6))
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 1.65e-6]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 1.65000000000000008e-6 < a

   1. Initial program 87.7%

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

   5. Applied rewrites 76.8%

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

   if -0.0115 < a < 1.65000000000000008e-6

   1. Initial program 75.8%

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 1.65e-6)))
   (fma (- t x) (/ y a) x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 1.65e-6)) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 1.65e-6))
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 1.65e-6]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 1.65000000000000008e-6 < a

   1. Initial program 87.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 89.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 70.5%

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

   if -0.0115 < a < 1.65000000000000008e-6

   1. Initial program 75.8%

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 1.65e-6)))
   (fma (/ (- t x) a) y x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 1.65e-6)) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 1.65e-6))
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 1.65e-6]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 1.65000000000000008e-6 < a

   1. Initial program 87.7%

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

   5. Applied rewrites 69.2%

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

   if -0.0115 < a < 1.65000000000000008e-6

   1. Initial program 75.8%

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0115) (not (<= a 3.1e-6)))
   (fma t (/ y a) x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0115) || !(a <= 3.1e-6)) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0115) || !(a <= 3.1e-6))
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0115], N[Not[LessEqual[a, 3.1e-6]], $MachinePrecision]], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0115 or 3.1e-6 < a

   1. Initial program 87.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 89.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 70.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 64.6%

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

   if -0.0115 < a < 3.1e-6

   1. Initial program 75.8%

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0115 \lor \neg \left(a \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.1 \cdot 10^{+21}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.1e+21) t (if (<= z 2.8e+87) (fma t (/ y a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.1e+21) {
		tmp = t;
	} else if (z <= 2.8e+87) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.1e+21)
		tmp = t;
	elseif (z <= 2.8e+87)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.1e+21], t, If[LessEqual[z, 2.8e+87], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1e21 or 2.80000000000000015e87 < z

   1. Initial program 68.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 54.1%

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

   if -8.1e21 < z < 2.80000000000000015e87

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{{\left(\sqrt[3]{a - z}\right)}^{2}}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\color{blue}{\left(\sqrt[3]{a - z}\right)}}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \color{blue}{\frac{t - x}{\sqrt[3]{a - z}}}, x\right) \]

      14. lower-cbrt.f64 92.9

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\color{blue}{\sqrt[3]{a - z}}}, x\right) \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}}, \frac{t - x}{\sqrt[3]{a - z}}, x\right)} \]

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 67.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 60.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.1 \cdot 10^{+21}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.1 \cdot 10^{+21}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.1e+21) t (if (<= z 9.5e+83) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.1e+21) {
		tmp = t;
	} else if (z <= 9.5e+83) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.1d+21)) then
        tmp = t
    else if (z <= 9.5d+83) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.1e+21) {
		tmp = t;
	} else if (z <= 9.5e+83) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.1e+21:
		tmp = t
	elif z <= 9.5e+83:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.1e+21)
		tmp = t;
	elseif (z <= 9.5e+83)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.1e+21)
		tmp = t;
	elseif (z <= 9.5e+83)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.1e+21], t, If[LessEqual[z, 9.5e+83], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1e21 or 9.5000000000000002e83 < z

   1. Initial program 68.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 54.1%

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

   if -8.1e21 < z < 9.5000000000000002e83

   1. Initial program 91.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 37.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.1 \cdot 10^{+21}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 27.5%

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

6. Final simplification 27.5%

   \[\leadsto t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))