Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Percentage Accurate: 85.0% → 99.3%

Time: 7.2s

Alternatives: 10

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{z - y}{\frac{a - z}{t}}\\ t_2 := t \cdot \left(y - z\right)\\ t_3 := \frac{t\_2}{a - z}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+231}:\\ \;\;\;\;x - \frac{t\_2}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- z y) (/ (- a z) t))))
        (t_2 (* t (- y z)))
        (t_3 (/ t_2 (- a z))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 1e+231) (- x (/ t_2 (- z a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z - y) / ((a - z) / t));
	double t_2 = t * (y - z);
	double t_3 = t_2 / (a - z);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= 1e+231) {
		tmp = x - (t_2 / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z - y) / ((a - z) / t));
	double t_2 = t * (y - z);
	double t_3 = t_2 / (a - z);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= 1e+231) {
		tmp = x - (t_2 / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - ((z - y) / ((a - z) / t))
	t_2 = t * (y - z)
	t_3 = t_2 / (a - z)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= 1e+231:
		tmp = x - (t_2 / (z - a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(z - y) / Float64(Float64(a - z) / t)))
	t_2 = Float64(t * Float64(y - z))
	t_3 = Float64(t_2 / Float64(a - z))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= 1e+231)
		tmp = Float64(x - Float64(t_2 / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - ((z - y) / ((a - z) / t));
	t_2 = t * (y - z);
	t_3 = t_2 / (a - z);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= 1e+231)
		tmp = x - (t_2 / (z - a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(z - y), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, 1e+231], N[(x - N[(t$95$2 / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0 or 1.0000000000000001e231 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 48.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 1.0000000000000001e231

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq -\infty:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t}}\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 10^{+231}:\\ \;\;\;\;x - \frac{t \cdot \left(y - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z - y\right) \cdot \frac{t}{z - a}\\ t_2 := t \cdot \left(y - z\right)\\ t_3 := \frac{t\_2}{a - z}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+231}:\\ \;\;\;\;x - \frac{t\_2}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z y) (/ t (- z a))))
        (t_2 (* t (- y z)))
        (t_3 (/ t_2 (- a z))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 1e+231) (- x (/ t_2 (- z a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) * (t / (z - a));
	double t_2 = t * (y - z);
	double t_3 = t_2 / (a - z);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= 1e+231) {
		tmp = x - (t_2 / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) * (t / (z - a));
	double t_2 = t * (y - z);
	double t_3 = t_2 / (a - z);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= 1e+231) {
		tmp = x - (t_2 / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - y) * (t / (z - a))
	t_2 = t * (y - z)
	t_3 = t_2 / (a - z)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= 1e+231:
		tmp = x - (t_2 / (z - a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) * Float64(t / Float64(z - a)))
	t_2 = Float64(t * Float64(y - z))
	t_3 = Float64(t_2 / Float64(a - z))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= 1e+231)
		tmp = Float64(x - Float64(t_2 / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - y) * (t / (z - a));
	t_2 = t * (y - z);
	t_3 = t_2 / (a - z);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= 1e+231)
		tmp = x - (t_2 / (z - a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, 1e+231], N[(x - N[(t$95$2 / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0 or 1.0000000000000001e231 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 48.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 1.0000000000000001e231

   1. Initial program 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq -\infty:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 10^{+231}:\\ \;\;\;\;x - \frac{t \cdot \left(y - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z - y\right) \cdot \frac{t}{z - a}\\ t_2 := \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z y) (/ t (- z a)))) (t_2 (/ (* t (- y z)) (- a z))))
   (if (<= t_2 -1e+24) t_1 (if (<= t_2 2e+74) (fma t (/ z (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) * (t / (z - a));
	double t_2 = (t * (y - z)) / (a - z);
	double tmp;
	if (t_2 <= -1e+24) {
		tmp = t_1;
	} else if (t_2 <= 2e+74) {
		tmp = fma(t, (z / (z - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) * Float64(t / Float64(z - a)))
	t_2 = Float64(Float64(t * Float64(y - z)) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -1e+24)
		tmp = t_1;
	elseif (t_2 <= 2e+74)
		tmp = fma(t, Float64(z / Float64(z - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+24], t$95$1, If[LessEqual[t$95$2, 2e+74], N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -9.9999999999999998e23 or 1.9999999999999999e74 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -9.9999999999999998e23 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 1.9999999999999999e74

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{-1}{\frac{\color{blue}{0 - \left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}{\left(y - z\right) \cdot t}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}{\left(y - z\right) \cdot t}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}{\left(y - z\right) \cdot t}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 99.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 90.5

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

   7. Applied rewrites 90.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{t \cdot y}{z - a}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* t y) (- z a)))))
   (if (<= y -3e+48) t_1 (if (<= y 2.7e+16) (fma t (/ z (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((t * y) / (z - a));
	double tmp;
	if (y <= -3e+48) {
		tmp = t_1;
	} else if (y <= 2.7e+16) {
		tmp = fma(t, (z / (z - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(t * y) / Float64(z - a)))
	tmp = 0.0
	if (y <= -3e+48)
		tmp = t_1;
	elseif (y <= 2.7e+16)
		tmp = fma(t, Float64(z / Float64(z - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(t * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+48], t$95$1, If[LessEqual[y, 2.7e+16], N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e48 or 2.7e16 < y

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3e48 < y < 2.7e16

   1. Initial program 86.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{-1}{\frac{\color{blue}{0 - \left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}{\left(y - z\right) \cdot t}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}{\left(y - z\right) \cdot t}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}{\left(y - z\right) \cdot t}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 86.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.5

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 92.1

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

   7. Applied rewrites 92.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+48}:\\ \;\;\;\;x - \frac{t \cdot y}{z - a}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 36000000000000:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ z (- z a)) x)))
   (if (<= z -1.25e-126)
     t_1
     (if (<= z 36000000000000.0) (fma (- y z) (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (z / (z - a)), x);
	double tmp;
	if (z <= -1.25e-126) {
		tmp = t_1;
	} else if (z <= 36000000000000.0) {
		tmp = fma((y - z), (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(z / Float64(z - a)), x)
	tmp = 0.0
	if (z <= -1.25e-126)
		tmp = t_1;
	elseif (z <= 36000000000000.0)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.25e-126], t$95$1, If[LessEqual[z, 36000000000000.0], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000001e-126 or 3.6e13 < z

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{-1}{\frac{\color{blue}{0 - \left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}{\left(y - z\right) \cdot t}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}{\left(y - z\right) \cdot t}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}{\left(y - z\right) \cdot t}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 81.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if -1.25000000000000001e-126 < z < 3.6e13

   1. Initial program 96.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 6: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{z}{z - a}, x\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 36000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ z (- z a)) x)))
   (if (<= z -5.6e-105)
     t_1
     (if (<= z 36000000000000.0) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (z / (z - a)), x);
	double tmp;
	if (z <= -5.6e-105) {
		tmp = t_1;
	} else if (z <= 36000000000000.0) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(z / Float64(z - a)), x)
	tmp = 0.0
	if (z <= -5.6e-105)
		tmp = t_1;
	elseif (z <= 36000000000000.0)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.6e-105], t$95$1, If[LessEqual[z, 36000000000000.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.6e-105 or 3.6e13 < z

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{-1}{\frac{\color{blue}{0 - \left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a - z\right)}}{\left(y - z\right) \cdot t}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}{\left(y - z\right) \cdot t}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{-1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}{\left(y - z\right) \cdot t}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}{\left(y - z\right) \cdot t}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 80.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 82.4

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

   7. Applied rewrites 82.4%

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

   if -5.6e-105 < z < 3.6e13

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 7: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-75}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 40000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.7e-75)
   (+ x t)
   (if (<= z 40000000000000.0) (fma (/ y a) t x) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.7e-75) {
		tmp = x + t;
	} else if (z <= 40000000000000.0) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.7e-75)
		tmp = Float64(x + t);
	elseif (z <= 40000000000000.0)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.7e-75], N[(x + t), $MachinePrecision], If[LessEqual[z, 40000000000000.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.69999999999999958e-75 or 4e13 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -7.69999999999999958e-75 < z < 4e13

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-75}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 40000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.2e+237) (* (/ y a) t) (+ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.2e+237) {
		tmp = (y / a) * t;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-1.2d+237)) then
        tmp = (y / a) * t
    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 (y <= -1.2e+237) {
		tmp = (y / a) * t;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.2e+237:
		tmp = (y / a) * t
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.2e+237)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.2e+237)
		tmp = (y / a) * t;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.2e+237], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], N[(x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1999999999999999e237

   1. Initial program 93.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.4%

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

   if -1.1999999999999999e237 < y

   1. Initial program 87.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+238}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -4.3e+238) (/ (* t y) a) (+ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -4.3e+238) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-4.3d+238)) 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 (y <= -4.3e+238) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -4.3e+238:
		tmp = (t * y) / a
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -4.3e+238)
		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 (y <= -4.3e+238)
		tmp = (t * y) / a;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -4.3e+238], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], N[(x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999983e238

   1. Initial program 93.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.0%

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

   if -4.29999999999999983e238 < y

   1. Initial program 87.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+238}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. lower-+.f64 60.1

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

5. Applied rewrites 60.1%

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

6. Final simplification 60.1%

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

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -10682974490174067/10000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 312887599100691/80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t)))))

  (+ x (/ (* (- y z) t) (- a z))))