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

Percentage Accurate: 68.5% → 85.3%

Time: 4.5s

Alternatives: 15

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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(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: 68.5% 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

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(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: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot \frac{a - y}{z} + t\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (- x) (/ (- a y) z)) t)))
   (if (<= z -1.1e+203)
     t_1
     (if (<= z 1.35e+95) (fma (- y z) (/ (- t x) (- a z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-x * ((a - y) / z)) + t;
	double tmp;
	if (z <= -1.1e+203) {
		tmp = t_1;
	} else if (z <= 1.35e+95) {
		tmp = fma((y - z), ((t - x) / (a - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-x) * Float64(Float64(a - y) / z)) + t)
	tmp = 0.0
	if (z <= -1.1e+203)
		tmp = t_1;
	elseif (z <= 1.35e+95)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / Float64(a - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.1e+203], t$95$1, If[LessEqual[z, 1.35e+95], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000002e203 or 1.35e95 < z

   1. Initial program 32.1%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if -1.10000000000000002e203 < z < 1.35e95

   1. Initial program 80.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 87.5

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

   3. Applied rewrites 87.5%

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

Alternative 2: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot \frac{a - y}{z} + t\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+111}:\\ \;\;\;\;\left(-\frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (- x) (/ (- a y) z)) t)))
   (if (<= z -1.12e+170)
     t_1
     (if (<= z -4e-140)
       (fma (- y z) (/ t (- a z)) x)
       (if (<= z 4.2e-75)
         (fma (- t x) (/ (- y z) a) x)
         (if (<= z 4.5e+111) (+ (- (/ (* (- t x) (- y a)) z)) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-x * ((a - y) / z)) + t;
	double tmp;
	if (z <= -1.12e+170) {
		tmp = t_1;
	} else if (z <= -4e-140) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (z <= 4.2e-75) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (z <= 4.5e+111) {
		tmp = -(((t - x) * (y - a)) / z) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-x) * Float64(Float64(a - y) / z)) + t)
	tmp = 0.0
	if (z <= -1.12e+170)
		tmp = t_1;
	elseif (z <= -4e-140)
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (z <= 4.2e-75)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (z <= 4.5e+111)
		tmp = Float64(Float64(-Float64(Float64(Float64(t - x) * Float64(y - a)) / z)) + t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.12e+170], t$95$1, If[LessEqual[z, -4e-140], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.2e-75], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.5e+111], N[((-N[(N[(N[(t - x), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1200000000000001e170 or 4.50000000000000001e111 < z

   1. Initial program 31.5%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -1.1200000000000001e170 < z < -3.9999999999999999e-140

   1. Initial program 73.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 85.0

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.6%

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

   if -3.9999999999999999e-140 < z < 4.2000000000000002e-75

   1. Initial program 91.2%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 85.1

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

   4. Applied rewrites 85.1%

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

   if 4.2000000000000002e-75 < z < 4.50000000000000001e111

   1. Initial program 74.2%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 52.3%

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

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot \frac{a - y}{z} + t\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+203}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (- x) (/ (- a y) z)) t)))
   (if (<= z -1.12e+170)
     t_1
     (if (<= z -4e-140)
       (fma (- y z) (/ t (- a z)) x)
       (if (<= z 3.6e-24)
         (fma (- t x) (/ (- y z) a) x)
         (if (<= z 1.35e+203) (- t (* (/ (- t x) z) y)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-x * ((a - y) / z)) + t;
	double tmp;
	if (z <= -1.12e+170) {
		tmp = t_1;
	} else if (z <= -4e-140) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (z <= 3.6e-24) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (z <= 1.35e+203) {
		tmp = t - (((t - x) / z) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-x) * Float64(Float64(a - y) / z)) + t)
	tmp = 0.0
	if (z <= -1.12e+170)
		tmp = t_1;
	elseif (z <= -4e-140)
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (z <= 3.6e-24)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (z <= 1.35e+203)
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.12e+170], t$95$1, If[LessEqual[z, -4e-140], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.6e-24], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+203], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1200000000000001e170 or 1.35e203 < z

   1. Initial program 27.5%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -1.1200000000000001e170 < z < -3.9999999999999999e-140

   1. Initial program 73.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. 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. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 85.0

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.6%

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

   if -3.9999999999999999e-140 < z < 3.6000000000000001e-24

   1. Initial program 90.7%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 82.8

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

   4. Applied rewrites 82.8%

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

   if 3.6000000000000001e-24 < z < 1.35e203

   1. Initial program 58.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 58.9

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

   7. Applied rewrites 58.9%

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

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{t - x}{z} \cdot y\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{a - y}{z} + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ (- t x) z) y))))
   (if (<= z -1.1e+33)
     t_1
     (if (<= z 3.6e-24)
       (fma (- t x) (/ (- y z) a) x)
       (if (<= z 1.35e+203) t_1 (+ (* (- x) (/ (- a y) z)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (((t - x) / z) * y);
	double tmp;
	if (z <= -1.1e+33) {
		tmp = t_1;
	} else if (z <= 3.6e-24) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (z <= 1.35e+203) {
		tmp = t_1;
	} else {
		tmp = (-x * ((a - y) / z)) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(Float64(t - x) / z) * y))
	tmp = 0.0
	if (z <= -1.1e+33)
		tmp = t_1;
	elseif (z <= 3.6e-24)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (z <= 1.35e+203)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(a - y) / z)) + t);
	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] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+33], t$95$1, If[LessEqual[z, 3.6e-24], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+203], t$95$1, N[(N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.09999999999999997e33 or 3.6000000000000001e-24 < z < 1.35e203

   1. Initial program 51.1%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 64.3

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

   7. Applied rewrites 64.3%

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

   if -1.09999999999999997e33 < z < 3.6000000000000001e-24

   1. Initial program 89.3%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if 1.35e203 < z

   1. Initial program 51.1%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 62.6

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

   7. Applied rewrites 62.6%

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

Alternative 5: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{t - x}{z} \cdot y\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{a - y}{z} + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ (- t x) z) y))))
   (if (<= z -7.6e+32)
     t_1
     (if (<= z 4.2e-75)
       (fma y (/ (- t x) a) x)
       (if (<= z 1.35e+203) t_1 (+ (* (- x) (/ (- a y) z)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (((t - x) / z) * y);
	double tmp;
	if (z <= -7.6e+32) {
		tmp = t_1;
	} else if (z <= 4.2e-75) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (z <= 1.35e+203) {
		tmp = t_1;
	} else {
		tmp = (-x * ((a - y) / z)) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(Float64(t - x) / z) * y))
	tmp = 0.0
	if (z <= -7.6e+32)
		tmp = t_1;
	elseif (z <= 4.2e-75)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (z <= 1.35e+203)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-x) * Float64(Float64(a - y) / z)) + t);
	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] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.6e+32], t$95$1, If[LessEqual[z, 4.2e-75], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+203], t$95$1, N[(N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6000000000000006e32 or 4.2000000000000002e-75 < z < 1.35e203

   1. Initial program 54.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if -7.6000000000000006e32 < z < 4.2000000000000002e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.8

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

   4. Applied rewrites 73.8%

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

   if 1.35e203 < z

   1. Initial program 54.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 59.8

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

   7. Applied rewrites 59.8%

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

Alternative 6: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{t - x}{z} \cdot y\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ (- t x) z) y))))
   (if (<= z -7.6e+32) t_1 (if (<= z 4.2e-75) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (((t - x) / z) * y);
	double tmp;
	if (z <= -7.6e+32) {
		tmp = t_1;
	} else if (z <= 4.2e-75) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(Float64(t - x) / z) * y))
	tmp = 0.0
	if (z <= -7.6e+32)
		tmp = t_1;
	elseif (z <= 4.2e-75)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.6e+32], t$95$1, If[LessEqual[z, 4.2e-75], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6000000000000006e32 or 4.2000000000000002e-75 < z

   1. Initial program 50.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -7.6000000000000006e32 < z < 4.2000000000000002e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 7: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z} + t\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* x (/ y z)) t)))
   (if (<= z -1.2e+33)
     t_1
     (if (<= z 4.2e-75)
       (fma y (/ (- t x) a) x)
       (if (<= z 2.9e+93) (* (/ (- x t) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (y / z)) + t;
	double tmp;
	if (z <= -1.2e+33) {
		tmp = t_1;
	} else if (z <= 4.2e-75) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (z <= 2.9e+93) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * Float64(y / z)) + t)
	tmp = 0.0
	if (z <= -1.2e+33)
		tmp = t_1;
	elseif (z <= 4.2e-75)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (z <= 2.9e+93)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.2e+33], t$95$1, If[LessEqual[z, 4.2e-75], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.9e+93], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e33 or 2.8999999999999998e93 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if -1.2e33 < z < 4.2000000000000002e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.8

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

   4. Applied rewrites 73.8%

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

   if 4.2000000000000002e-75 < z < 2.8999999999999998e93

   1. Initial program 76.1%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 8: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z} + t\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* x (/ y z)) t)))
   (if (<= z -1.2e+33)
     t_1
     (if (<= z 4.2e-75)
       (fma t (/ y a) x)
       (if (<= z 2.9e+93) (* (/ (- x t) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (y / z)) + t;
	double tmp;
	if (z <= -1.2e+33) {
		tmp = t_1;
	} else if (z <= 4.2e-75) {
		tmp = fma(t, (y / a), x);
	} else if (z <= 2.9e+93) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * Float64(y / z)) + t)
	tmp = 0.0
	if (z <= -1.2e+33)
		tmp = t_1;
	elseif (z <= 4.2e-75)
		tmp = fma(t, Float64(y / a), x);
	elseif (z <= 2.9e+93)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.2e+33], t$95$1, If[LessEqual[z, 4.2e-75], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.9e+93], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e33 or 2.8999999999999998e93 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if -1.2e33 < z < 4.2000000000000002e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 75.9

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.9%

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

   if 4.2000000000000002e-75 < z < 2.8999999999999998e93

   1. Initial program 76.1%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 9: 53.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35)
   t
   (if (<= z 4.2e-75)
     (fma t (/ y a) x)
     (if (<= z 5e+93) (* (/ (- x t) z) y) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 4.2e-75) {
		tmp = fma(t, (y / a), x);
	} else if (z <= 5e+93) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 4.2e-75)
		tmp = fma(t, Float64(y / a), x);
	elseif (z <= 5e+93)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 4.2e-75], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5e+93], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999998e35 or 5.0000000000000001e93 < z

   1. Initial program 40.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.7%

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

   if -2.2999999999999998e35 < z < 4.2000000000000002e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.7%

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

   if 4.2000000000000002e-75 < z < 5.0000000000000001e93

   1. Initial program 76.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 10: 52.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35) t (if (<= z 4.3e+83) (fma t (/ y a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 4.3e+83) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 4.3e+83)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 4.3e+83], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999998e35 or 4.3e83 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.2%

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

   if -2.2999999999999998e35 < z < 4.3e83

   1. Initial program 87.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.6

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

   4. Applied rewrites 67.6%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 57.9%

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

Alternative 11: 52.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35) t (if (<= z 4e+83) (fma y (/ t a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 4e+83) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 4e+83)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 4e+83], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999998e35 or 4.00000000000000012e83 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.2%

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

   if -2.2999999999999998e35 < z < 4.00000000000000012e83

   1. Initial program 87.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 56.0%

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

Alternative 12: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35)
   t
   (if (<= z 2.6e-148) x (if (<= z 7.5e+79) (* (/ x z) y) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 2.6e-148) {
		tmp = x;
	} else if (z <= 7.5e+79) {
		tmp = (x / z) * y;
	} 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 <= (-2.3d+35)) then
        tmp = t
    else if (z <= 2.6d-148) then
        tmp = x
    else if (z <= 7.5d+79) then
        tmp = (x / z) * y
    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 <= -2.3e+35) {
		tmp = t;
	} else if (z <= 2.6e-148) {
		tmp = x;
	} else if (z <= 7.5e+79) {
		tmp = (x / z) * y;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+35:
		tmp = t
	elif z <= 2.6e-148:
		tmp = x
	elif z <= 7.5e+79:
		tmp = (x / z) * y
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 2.6e-148)
		tmp = x;
	elseif (z <= 7.5e+79)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 2.6e-148)
		tmp = x;
	elseif (z <= 7.5e+79)
		tmp = (x / z) * y;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 2.6e-148], x, If[LessEqual[z, 7.5e+79], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999998e35 or 7.49999999999999967e79 < z

   1. Initial program 41.0%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.0%

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

   if -2.2999999999999998e35 < z < 2.60000000000000008e-148

   1. Initial program 89.7%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 34.6%

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

   if 2.60000000000000008e-148 < z < 7.49999999999999967e79

   1. Initial program 81.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 28.7

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 17.9%

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

Alternative 13: 36.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35)
   t
   (if (<= z 2.6e-148) x (if (<= z 7.5e+79) (/ (* x y) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 2.6e-148) {
		tmp = x;
	} else if (z <= 7.5e+79) {
		tmp = (x * y) / z;
	} 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 <= (-2.3d+35)) then
        tmp = t
    else if (z <= 2.6d-148) then
        tmp = x
    else if (z <= 7.5d+79) then
        tmp = (x * y) / z
    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 <= -2.3e+35) {
		tmp = t;
	} else if (z <= 2.6e-148) {
		tmp = x;
	} else if (z <= 7.5e+79) {
		tmp = (x * y) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+35:
		tmp = t
	elif z <= 2.6e-148:
		tmp = x
	elif z <= 7.5e+79:
		tmp = (x * y) / z
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 2.6e-148)
		tmp = x;
	elseif (z <= 7.5e+79)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 2.6e-148)
		tmp = x;
	elseif (z <= 7.5e+79)
		tmp = (x * y) / z;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 2.6e-148], x, If[LessEqual[z, 7.5e+79], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999998e35 or 7.49999999999999967e79 < z

   1. Initial program 41.0%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.0%

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

   if -2.2999999999999998e35 < z < 2.60000000000000008e-148

   1. Initial program 89.7%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 34.6%

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

   if 2.60000000000000008e-148 < z < 7.49999999999999967e79

   1. Initial program 81.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 28.7

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 18.5

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

   10. Applied rewrites 18.5%

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

Alternative 14: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+35}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 22000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+35) t (if (<= z 22000000000000.0) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+35) {
		tmp = t;
	} else if (z <= 22000000000000.0) {
		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 <= (-2.3d+35)) then
        tmp = t
    else if (z <= 22000000000000.0d0) 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 <= -2.3e+35) {
		tmp = t;
	} else if (z <= 22000000000000.0) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+35:
		tmp = t
	elif z <= 22000000000000.0:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+35)
		tmp = t;
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+35], t, If[LessEqual[z, 22000000000000.0], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999998e35 or 2.2e13 < z

   1. Initial program 44.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 44.4%

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

   if -2.2999999999999998e35 < z < 2.2e13

   1. Initial program 89.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 33.6%

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

Alternative 15: 25.1% accurate, 17.9× 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 68.5%

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

2. Taylor expanded in z around inf

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

4. Applied rewrites 25.1%

   \[\leadsto \color{blue}{t} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))