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

Percentage Accurate: 69.2% → 95.1%

Time: 6.3s

Alternatives: 13

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: 69.2% 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: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - z}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t - x, \mathsf{fma}\left(t\_1, t - x, x\right)\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t \cdot \left(z - y\right)}{a - z}, x \cdot \left(\frac{a}{a - z} - t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a z))))
   (if (<= t -4.2e+85)
     (fma (/ z (- z a)) (- t x) (fma t_1 (- t x) x))
     (if (<= t 3.1e+55)
       (fma -1.0 (/ (* t (- z y)) (- a z)) (* x (- (/ a (- a z)) t_1)))
       (fma (/ (- z y) (- z a)) (- t x) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - z);
	double tmp;
	if (t <= -4.2e+85) {
		tmp = fma((z / (z - a)), (t - x), fma(t_1, (t - x), x));
	} else if (t <= 3.1e+55) {
		tmp = fma(-1.0, ((t * (z - y)) / (a - z)), (x * ((a / (a - z)) - t_1)));
	} else {
		tmp = fma(((z - y) / (z - a)), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - z))
	tmp = 0.0
	if (t <= -4.2e+85)
		tmp = fma(Float64(z / Float64(z - a)), Float64(t - x), fma(t_1, Float64(t - x), x));
	elseif (t <= 3.1e+55)
		tmp = fma(-1.0, Float64(Float64(t * Float64(z - y)) / Float64(a - z)), Float64(x * Float64(Float64(a / Float64(a - z)) - t_1)));
	else
		tmp = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.2e+85], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + N[(t$95$1 * N[(t - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e+55], N[(-1.0 * N[(N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(a / N[(a - z), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000002e85

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   if -4.2000000000000002e85 < t < 3.09999999999999994e55

   1. Initial program 69.2%

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

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 63.6

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

   3. Applied rewrites 63.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 82.8

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

   6. Applied rewrites 82.8%

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

   if 3.09999999999999994e55 < t

   1. Initial program 69.2%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 84.8

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

   3. Applied rewrites 84.8%

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

Alternative 2: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(t - x\right), y - z, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t \cdot \left(z - y\right), x \cdot \left(a - y\right)\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t - x, \mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 -5e+60)
     (fma (* (/ 1.0 (- a z)) (- t x)) (- y z) x)
     (if (<= t_1 0.0)
       (/ (fma -1.0 (* t (- z y)) (* x (- a y))) (- a z))
       (fma (/ z (- z a)) (- t x) (fma (/ y (- a z)) (- t x) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -5e+60) {
		tmp = fma(((1.0 / (a - z)) * (t - x)), (y - z), x);
	} else if (t_1 <= 0.0) {
		tmp = fma(-1.0, (t * (z - y)), (x * (a - y))) / (a - z);
	} else {
		tmp = fma((z / (z - a)), (t - x), fma((y / (a - z)), (t - x), x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -5e+60)
		tmp = fma(Float64(Float64(1.0 / Float64(a - z)) * Float64(t - x)), Float64(y - z), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(-1.0, Float64(t * Float64(z - y)), Float64(x * Float64(a - y))) / Float64(a - z));
	else
		tmp = fma(Float64(z / Float64(z - a)), Float64(t - x), fma(Float64(y / Float64(a - z)), Float64(t - x), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+60], N[(N[(N[(1.0 / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -4.99999999999999975e60

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 80.5

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

   3. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 80.4

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

   5. Applied rewrites 80.4%

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

   if -4.99999999999999975e60 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 69.2%

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

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 63.6

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

   3. Applied rewrites 63.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 74.6

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

   6. Applied rewrites 74.6%

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

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

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

Alternative 3: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(t - x\right), y - z, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t \cdot \left(z - y\right), x \cdot \left(a - y\right)\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 -5e+60)
     (fma (* (/ 1.0 (- a z)) (- t x)) (- y z) x)
     (if (<= t_1 0.0)
       (/ (fma -1.0 (* t (- z y)) (* x (- a y))) (- a z))
       (fma (/ (- z y) (- z a)) (- t x) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -5e+60) {
		tmp = fma(((1.0 / (a - z)) * (t - x)), (y - z), x);
	} else if (t_1 <= 0.0) {
		tmp = fma(-1.0, (t * (z - y)), (x * (a - y))) / (a - z);
	} else {
		tmp = fma(((z - y) / (z - a)), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -5e+60)
		tmp = fma(Float64(Float64(1.0 / Float64(a - z)) * Float64(t - x)), Float64(y - z), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(-1.0, Float64(t * Float64(z - y)), Float64(x * Float64(a - y))) / Float64(a - z));
	else
		tmp = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+60], N[(N[(N[(1.0 / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -4.99999999999999975e60

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 80.5

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

   3. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 80.4

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

   5. Applied rewrites 80.4%

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

   if -4.99999999999999975e60 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 69.2%

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

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

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

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

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

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 63.6

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

   3. Applied rewrites 63.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 74.6

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

   6. Applied rewrites 74.6%

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

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

   1. Initial program 69.2%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 84.8

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

   3. Applied rewrites 84.8%

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

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -8e+168)
     t_1
     (if (<= z 4.5e+163) (fma (/ (- x t) (- z a)) (- y z) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -8e+168) {
		tmp = t_1;
	} else if (z <= 4.5e+163) {
		tmp = fma(((x - t) / (z - a)), (y - z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -8e+168)
		tmp = t_1;
	elseif (z <= 4.5e+163)
		tmp = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -8e+168], t$95$1, If[LessEqual[z, 4.5e+163], N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.9999999999999995e168 or 4.49999999999999988e163 < z

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -7.9999999999999995e168 < z < 4.49999999999999988e163

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 80.5

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

   3. Applied rewrites 80.5%

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

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e+183)
   (fma (- x t) (/ y z) t)
   (fma (/ (- z y) (- z a)) (- t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+183) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma(((z - y) / (z - a)), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e+183)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e+183], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e183

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -1.05e183 < z

   1. Initial program 69.2%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 84.8

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

   3. Applied rewrites 84.8%

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

Alternative 6: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{a}, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -1.2e+23)
     t_1
     (if (<= z 2.7e+30) (fma (/ (- x t) a) (- z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -1.2e+23) {
		tmp = t_1;
	} else if (z <= 2.7e+30) {
		tmp = fma(((x - t) / a), (z - y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1.2e+23)
		tmp = t_1;
	elseif (z <= 2.7e+30)
		tmp = fma(Float64(Float64(x - t) / a), Float64(z - y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.2e+23], t$95$1, If[LessEqual[z, 2.7e+30], N[(N[(N[(x - t), $MachinePrecision] / a), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e23 or 2.6999999999999999e30 < z

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -1.2e23 < z < 2.6999999999999999e30

   1. Initial program 69.2%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 84.8

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

   3. Applied rewrites 84.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

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

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. mult-flip N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 51.8

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

   8. Applied rewrites 51.8%

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

Alternative 7: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0098:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -1e+22) t_1 (if (<= z 0.0098) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -1e+22) {
		tmp = t_1;
	} else if (z <= 0.0098) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1e+22)
		tmp = t_1;
	elseif (z <= 0.0098)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1e+22], t$95$1, If[LessEqual[z, 0.0098], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e22 or 0.0097999999999999997 < z

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -1e22 < z < 0.0097999999999999997

   1. Initial program 69.2%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 84.8

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.8

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

   6. Applied rewrites 48.8%

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

Alternative 8: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -9e-32) t_1 (if (<= z 6.5e-31) (* y (/ (- t x) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -9e-32) {
		tmp = t_1;
	} else if (z <= 6.5e-31) {
		tmp = y * ((t - x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -9e-32)
		tmp = t_1;
	elseif (z <= 6.5e-31)
		tmp = Float64(y * Float64(Float64(t - x) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -9e-32], t$95$1, If[LessEqual[z, 6.5e-31], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000009e-32 or 6.49999999999999967e-31 < z

   1. Initial program 69.2%

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

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

      3. remove-double-neg N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. div-add N/A

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

      16. remove-double-neg N/A

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

      17. associate-+l+ N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if -9.00000000000000009e-32 < z < 6.49999999999999967e-31

   1. Initial program 69.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 25.8

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

   7. Applied rewrites 25.8%

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

Alternative 9: 40.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-28}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e-28) (+ x t) (if (<= z 5.1e-20) (* y (/ (- t x) a)) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-28) {
		tmp = x + t;
	} else if (z <= 5.1e-20) {
		tmp = y * ((t - x) / a);
	} else {
		tmp = x + 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 <= (-1.4d-28)) then
        tmp = x + t
    else if (z <= 5.1d-20) then
        tmp = y * ((t - x) / a)
    else
        tmp = x + 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 <= -1.4e-28) {
		tmp = x + t;
	} else if (z <= 5.1e-20) {
		tmp = y * ((t - x) / a);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e-28:
		tmp = x + t
	elif z <= 5.1e-20:
		tmp = y * ((t - x) / a)
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e-28)
		tmp = Float64(x + t);
	elseif (z <= 5.1e-20)
		tmp = Float64(y * Float64(Float64(t - x) / a));
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e-28)
		tmp = x + t;
	elseif (z <= 5.1e-20)
		tmp = y * ((t - x) / a);
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e-28], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.1e-20], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e-28 or 5.10000000000000019e-20 < z

   1. Initial program 69.2%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.7%

      \[\leadsto x + t \]

   if -1.3999999999999999e-28 < z < 5.10000000000000019e-20

   1. Initial program 69.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 25.8

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

   7. Applied rewrites 25.8%

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

Alternative 10: 35.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-28}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e-28) (+ x t) (if (<= z 5.5e-20) (* y (/ t (- a z))) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e-28) {
		tmp = x + t;
	} else if (z <= 5.5e-20) {
		tmp = y * (t / (a - z));
	} else {
		tmp = x + 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 <= (-1.4d-28)) then
        tmp = x + t
    else if (z <= 5.5d-20) then
        tmp = y * (t / (a - z))
    else
        tmp = x + 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 <= -1.4e-28) {
		tmp = x + t;
	} else if (z <= 5.5e-20) {
		tmp = y * (t / (a - z));
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e-28:
		tmp = x + t
	elif z <= 5.5e-20:
		tmp = y * (t / (a - z))
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e-28)
		tmp = Float64(x + t);
	elseif (z <= 5.5e-20)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e-28)
		tmp = x + t;
	elseif (z <= 5.5e-20)
		tmp = y * (t / (a - z));
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e-28], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.5e-20], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e-28 or 5.4999999999999996e-20 < z

   1. Initial program 69.2%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.7%

      \[\leadsto x + t \]

   if -1.3999999999999999e-28 < z < 5.4999999999999996e-20

   1. Initial program 69.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 23.1

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

   7. Applied rewrites 23.1%

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

Alternative 11: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-138}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-138) (+ x t) (if (<= z 5.1e-20) (/ (* t y) (- a z)) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-138) {
		tmp = x + t;
	} else if (z <= 5.1e-20) {
		tmp = (t * y) / (a - z);
	} else {
		tmp = x + 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 <= (-1d-138)) then
        tmp = x + t
    else if (z <= 5.1d-20) then
        tmp = (t * y) / (a - z)
    else
        tmp = x + 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 <= -1e-138) {
		tmp = x + t;
	} else if (z <= 5.1e-20) {
		tmp = (t * y) / (a - z);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-138:
		tmp = x + t
	elif z <= 5.1e-20:
		tmp = (t * y) / (a - z)
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-138)
		tmp = Float64(x + t);
	elseif (z <= 5.1e-20)
		tmp = Float64(Float64(t * y) / Float64(a - z));
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e-138)
		tmp = x + t;
	elseif (z <= 5.1e-20)
		tmp = (t * y) / (a - z);
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-138], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.1e-20], N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000007e-138 or 5.10000000000000019e-20 < z

   1. Initial program 69.2%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.7%

      \[\leadsto x + t \]

   if -1.00000000000000007e-138 < z < 5.10000000000000019e-20

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.8

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 21.3

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

   7. Applied rewrites 21.3%

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

Alternative 12: 35.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-178}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e-178) (+ x t) (if (<= z 5.5e-167) (/ (* t y) a) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-178) {
		tmp = x + t;
	} else if (z <= 5.5e-167) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.5d-178)) then
        tmp = x + t
    else if (z <= 5.5d-167) then
        tmp = (t * y) / a
    else
        tmp = x + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-178) {
		tmp = x + t;
	} else if (z <= 5.5e-167) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.5e-178:
		tmp = x + t
	elif z <= 5.5e-167:
		tmp = (t * y) / a
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e-178)
		tmp = Float64(x + t);
	elseif (z <= 5.5e-167)
		tmp = Float64(Float64(t * y) / a);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.5e-178)
		tmp = x + t;
	elseif (z <= 5.5e-167)
		tmp = (t * y) / a;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e-178], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.5e-167], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000001e-178 or 5.5000000000000003e-167 < z

   1. Initial program 69.2%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.7%

      \[\leadsto x + t \]

   if -8.5000000000000001e-178 < z < 5.5000000000000003e-167

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.8

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.6

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

   7. Applied rewrites 16.6%

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

Alternative 13: 33.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 + t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 19.8

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

4. Applied rewrites 19.8%

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

5. Taylor expanded in x around 0

   \[\leadsto x + t \]
6. Step-by-step derivation

7. Applied rewrites 33.7%

   \[\leadsto x + t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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))))